Examples of genuine QM abelian surfaces which are modular
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Abstract
Let K be an imaginary quadratic field. Modular forms for GL(2) over K are known as Bianchi modular forms. Standard modularity conjectures assert that every weight 2 rational Bianchi newform has either an associated elliptic curve over K or an associated abelian surface with quaternionic multiplication over K. We give explicit evidence in the way of examples to support this conjecture in the latter case. Furthermore, the quaternionic surfaces given correspond to genuine Bianchi newforms, which answers a question posed by Cremona as to whether this phenomenon can happen.
Keywords
Modularity QM abelian surface Bianchi FaltingsSerreMathematics Subject Classification
11F32 11G10 11F03 11F801 Introduction
Let K be an imaginary quadratic field. A simple abelian surface over K whose algebra of Kendomorphisms is an indefinite quaternion algebra over \(\mathbb {Q}\) is commonly known as a QM abelian surface, or just QM surface. These are also often referred to false elliptic curves, coined by Serre in the 1970s [1] based on the observation that such a surface is isogenous to the square of an elliptic curve modulo every prime of good reduction [30, Lemma 6].
It is wellknown that one can obtain QM surfaces over K by base changing suitable abelian surfaces over \(\mathbb {Q}\). Accordingly, let us call a QM surface over K genuine if it is not the twist of basechange to K of an abelian surface over \(\mathbb {Q}\). Motivated by the conjectural connections with Bianchi modular forms, in 1992 Cremona asked whether genuine QM surfaces over imaginary quadratic fields should exist (see Question 1.3). We answer this question in the positive by providing explicit genus 2 curves whose Jacobians are genuine QM surfaces. To the best of our knowledge these are the first such examples in the literature. Furthermore, by carrying out a detailed analysis of the associated Galois representations and applying the Faltings–Serre–Livné criterion, we prove the modularity of these QM surfaces. The main result of the present article is as follows:
Theorem 1.1
 1.
\(C_1: y^2= x^6 + 4ix^5 + (2i  6)x^4 + (i + 7)x^3 + (8i  9)x^2  10ix + 4i + 3\),
Bianchi newform: 2.0.4.134225.3a;
 2.
\(C_2 : y^2=x^6 + (2\sqrt{3}  10)x^5 + (10\sqrt{3} + 30)x^4 + (8\sqrt{3}  32)x^3\)
\(+ (4\sqrt{3} + 16)x^2 + (16\sqrt{3}  12)x  4\sqrt{3} + 16\),
Bianchi newform: 2.0.3.161009.1a;
 3.
\(C_3: y^2=(104\sqrt{3}  75)x^6 + (528\sqrt{3} + 456)x^4 + (500\sqrt{3} + 1044)x^3\)
\(+ (1038\sqrt{3} + 2706)x^2 + (1158\sqrt{3} + 342)x  612\sqrt{3}  1800\),
Bianchi newform: 2.0.3.167081.3a;
 4.
\(C_4 : y^2 = x^6  2\sqrt{3}x^5 + (2\sqrt{3}  3)x^4 + 1/3(2\sqrt{3} + 54)x^3\)
\(+ (20\sqrt{3} + 3)x^2 + (8\sqrt{3}  30)x + 4\sqrt{3}  11\),
Bianchi newform: 2.0.3.1123201.1b.
Proof
See Sect. 4. \(\square \)
Over the rationals there is a celebrated result that establishes a connection between elliptic curves over \(\mathbb {Q}\) and classical newforms of weight 2. Extending this to number fields is an important aspect of the Langlands programme. However, in the case when the number field is totally complex the correspondence needs to be modified to include QM surfaces. This was first observed by Deligne in a letter to Mennicke in 1979 [12] for imaginary quadratic fields. The construction is detailed in [15] and is illustrated with an explicit example.
Thus the details of modularity for QM surfaces are different from the case of \(\hbox {GL}_2\)type. Let us be completely explicit with the following conjecture [8, 29].
Conjecture 1.2
 1.
Let f be a Bianchi newform over K of weight 2 and level \(\varGamma _0(\mathfrak {n})\) with rational Hecke eigenvalues. Then there is either an elliptic curve E / K without CM by K of conductor \(\mathfrak {n}\) such that \(L(E/K,s)=L(f,s)\) or there is a QM surface A / K of conductor \(\mathfrak {n}^2\) such that \(L(A/K,s)= L(f,s)^2\).
 2.
Conversely, if E / K is an elliptic curve without CM by K of conductor \(\mathfrak {n}\) then there is an f as above such that \(L(E,s)=L(f,s)\). Moreover, if A / K is a QM surface of conductor \(\mathfrak {n}^2\) then there is an f as above such that \(L(A,s)= L(f,s)^2\).
Let f be a classical newform of weight 2 with a real quadratic Hecke eigenvalue field \(K_f= \mathbb {Q}(\{ a_i\})\) and denote \(<\sigma >=\text {Gal}(K_f / \mathbb {Q})\). We say that f has an inner twist if \(f^\sigma = f \otimes \chi _K\) where \(\chi _K\) is the quadratic Dirichlet character associated to some imaginary quadratic field K. It follows that f and \(f^\sigma \) must base change to the same Bianchi newform over K. The term genuine is used for newforms that are not (the twist of) basechange of a classical newform. For more background on Bianchi newforms see [11, §2].
From a geometric point of view, let \(A/\mathbb {Q}\) be the abelian surface of \(\hbox {GL}_2\)type corresponding to the newforms \(f, f^\sigma \) with an inner twist via \(L(A/\mathbb {Q},s)=L(f,s)L(f^\sigma ,s)\). If the basechange surface \(A \otimes _{\mathbb {Q}} K\) remains simple then it is necessarily a QM surface and \(L(A/K,s) = L(F,s)^2\), where F is the induction from \(\mathbb {Q}\) to K of f. This motivates the following question (see [10, Question 1’] and also [13, Conjecture 1]).
Question 1.3
If f is a rational weight 2 Bianchi newform over K which is genuine, does f have an associated elliptic curve over K?
Given the modular correspondence above, we could rephrase this question to ask whether all QM surfaces arise from a \(\hbox {GL}_2\)type surface over \(\mathbb {Q}\). The genus 2 curves given in Theorem 4.4 answer this question and say that such a newform f does not necessarily have to correspond to an elliptic curve.
The genuine QM surfaces we present also have an interesting connection to the Paramodularity Conjecture. Recall that the Paramodularity Conjecture posits a correspondence between abelian surfaces \(A/\mathbb {Q}\) with \(\text {End}_\mathbb {Q}(A) \otimes \mathbb {Q}\simeq \mathbb {Q}\) and genus 2 paramodular rational Siegel newforms of weight 2 that are not Gritsenko lifts [6, Conjecture 1.1]. It has been recently pointed out by F. Calegari et al. [5, §10] that the conjectural correspondence needs to include abelian 4folds \(B/\mathbb {Q}\) with \(\text {End}_\mathbb {Q}(B) \otimes \mathbb {Q}\) an indefinite nonsplit quaternion algebra over \(\mathbb {Q}\) (see the amended version in [6, §8]) . This can be illustrated using our genuine QM surfaces.
Let C / K be any of the four curves given in Theorem 4.4. Define A / K to be the QM surface given by taking the Jacobian of C / K with \(\text {End}_K(A) \otimes \mathbb {Q}\simeq D/\mathbb {Q}\) an indefinite nonsplit quaternion algebra. Then the Weil restriction \(B=\text {Res}_{K/\mathbb {Q}}(A)\) of A from K to \(\mathbb {Q}\) is a simple abelian 4fold such that \(\text {End}_\mathbb {Q}(B) \otimes \mathbb {Q}\simeq D/\mathbb {Q}\). We prove that there is a genuine rational weight 2 Bianchi newform f over K such that \(L(A/K,s)=L(f,s)^2\). Now let F be the genus 2 paramodular rational Siegel newform of weight 2 that is the theta lift of f. It now follows from the properties of Weil restriction [24] and theta lifting [3] that \(L(B/\mathbb {Q},s)=L(A/K,s)=L(f,s)^2=L(F,s)^2\).
In analogy to the case of QM surfaces, at any prime p unramified in D the 8dimensional padic Tate module of \(B/\mathbb {Q}\) splits as the square of a 4dimensional submodule[7, §7]. Then the 4dimensional padic Galois representation has similar arithmetic to one that arises from an abelian surface over \(\mathbb {Q}\) with trivial endomorphisms. Indeed, our example above shows that via the representation afforded by the submodule, \(B/\mathbb {Q}\) corresponds to a Siegel newform of the type considered in the Paramodularity Conjecture.
The article will be laid out as following: in Sect. 2 we outline how these genus 2 curves were found and in Sect. 3 we discuss some arithmetic properties of the attached Galois representation in the case where \(\ell \) divides the discriminant of the acting quaternion algebra. Then Sect. 4 will be dedicated to showing how the Faltings–Serre–Livné criterion can be applied in order to prove that the examples given are modular. The final section lists the examples and contains further details of interest about them.
2 Rational points on Shimura curves
In this section we outline how the genus 2 curves in Theorem 4.4 were found. Let A be a geometrically simple abelian surface defined over an imaginary quadratic field K. Define the endomorphism rings \(\hbox {End}_K(A)\) and \(\hbox {End}_{\overline{K}}(A)\) to be the endomorphisms of A which are defined over K and \(\overline{K}\) respectively. As in the introduction, we use the convention that A has quaternionic multiplication, or QM for short, if \(\hbox {End}_{{K}}(A)\) is an order \({\mathscr {O}}\) in an indefinite quaternion algebra \(B_D\) over \(\mathbb {Q}\). We say that A has potential QM if the action of \({\mathscr {O}}\) is defined over some extension of K. The notation \(B_D\) is used for the unique quaternion algebra of discriminant D up to isomorphism. Note that \(B_D\) must be nonsplit because A is simple. For the remainder of the article let \({\mathscr {O}}\) be a maximal order of \(B_D\).
Families of QM surfaces have been constructed by Hashimoto et al. (see [17]) for quaternion algebras of discriminant 6 and 10. Testing numerically it would seem these give rise to surfaces which are all (a twist of) basechange. So we instead utilise two families given by Baba and Granath [2] that have been derived from the moduli space.
Given the order \({\mathscr {O}}\), the set of norm 1 elements is denoted by \({\mathscr {O}}^1\). These act as isometries on the upper half plane \(\mathscr {H}_2\) via an embedding \({\mathscr {O}}\hookrightarrow M_2(\mathbb {R})\) and the resulting quotient \(X_D = \mathscr {H}_2 / {\mathscr {O}}^1\) is a moduli space for abelian surfaces with quaternionic multiplication by \({\mathscr {O}}\) [2]. It is well known that these are compact Riemann surfaces called Shimura curves and they admit a model defined over \(\mathbb {Q}\). In particular, these are \(X_6: X^2+ 3Y^2 + Z^2 = 0\) and \(X_{10}: X^2+ 2Y^2 + Z^2 = 0\).
So let us suppose that \(K \hookrightarrow B_D\). Once we have the curve \(C_j\) defined over \(K(\sqrt{6})\) we wish to find an isomorphic curve defined over K. Using MAGMA it is possible to take IgusaClebsch invariants and then create a model defined over K with the same IgusaClebsch invariants. This then allows us to test whether the curve is a twist of basechange. It can be easily shown that a curve is genuine if the Euler polynomials at a pair of conjugate primes are not the same (up to twists). If it is indeed genuine then we endeavour to find a smaller model for the curve.
The size of the level places a limitation on whether a Bianchi modular form can be computed. Hence we try to find surfaces with as small a conductor as possible. With QM surfaces it can be a challenge to find examples with small conductor (see [6, Section 8]).
It is necessary to know the conductor exactly since we wish to find the conjecturally associated Bianchi newform. The odd part of the conductor can be found using MAGMA. Computing the even part has recently been made possible using machinery developed in [14]. The support of the ideal generated by the discriminant of a genus 2 hyperelliptic curve contains the support of the conductor of its Jacobian and the inclusion can in fact be strict. This phenomenon arises especially when one works with curves that have very large coefficients.
The curves in Theorem 4.4 were then found by parameterising the conic \(X_6(K)\) and conducting a large search with varying jvalue. To control the support of the conductor and stay away from large primes we used the proposition below. Once suitable curves were discovered, minimal models were found using as yet unpublished code by L. Dembélé.
Proposition 2.1
Let \(C_j/k\) be a genus 2 curve as above. Then \(C_j\) has potentially good reduction at a prime \({\mathfrak {p}}\not \mid 6\) if and only if \(\nu _{\mathfrak {p}}(j)=0\).
Proof
See [2, Proposition 3.19]. \(\square \)
3 Galois representations attached to QM surfaces
In this section we describe the image of the Galois representation attached to a QM surface when the prime \(\ell \) divides the discriminant of the quaternion algebra. For a brief overview on the arithmetic of quaternion algebras see [23, Ch. 2].
Let A / K be a QM surface with \({\mathscr {O}}\hookrightarrow \text {End}_K(A)\) a maximal order in the quaternion algebra \(B/\mathbb {Q}\) and denote by \(\sigma _{\ell } : G_K \rightarrow GL_4 (\mathbb {Z}_\ell )\) the representation coming from the \(\ell \)adic Tate module \(T_\ell A= \displaystyle \lim _{\leftarrow n} \ A[\ell ^n]\).
Under the identification (1) and projecting as in the commutative diagram the image of \(\overline{\rho }_{\ell }\) will lie in \(\hbox {GL}_2(\mathbb {F}_{{\ell }^2})\). Furthermore, up to conjugation it can be assumed that the image of \(\overline{\rho }_{\ell }\) is contained in \(\hbox {GL}_2(\mathbb {F}_{\ell })\) by [19, Lemma 3.1]. Specifically, it will be contained in the nonsplit Cartan subgroup of \(\hbox {GL}_2(\mathbb {F}_{\ell })\), which is the unique cyclic subgroup of order \(\ell ^21\).
Theorem 3.1
Proof
Remark 3.2
If f is a Bianchi newform which corresponds to a QM surface with quaternion algebra \(B_D\), then the residual representation attached to f has cyclic image at the primes dividing the discriminant D.
Given a Bianchi newform with rational coefficients, it would be desirable to have a criterion which determines whether f should correspond to an elliptic curve or a QM surface. The above gives a necessary condition for f to correspond to a QM surface. We wish to know whether a sufficient condition also exists and if so whether it can be determined from computing the trace of Frobenius for a finite set of primes.
4 Proof of modularity
Let \(f \in S_2(\varGamma _0({\mathfrak {p}}_{13,1}^2 \cdot {\mathfrak {p}}_{19,1}^2))\) be the genuine Bianchi newform which is listed on the LMFDB database with label 2.0.3.161009.1a. We will show that f is modular to A. By the work of [16, 28] and more recently [4, 25], we can associate an \(\ell \)adic Galois representation \(\rho _{f,\ell }:\mathrm{Gal}(\overline{K}/K) \rightarrow \mathrm{GL}_2(\overline{\mathbb {Q}}_{\ell })\) to f such that \(L(f,s) = L(\rho _{f,\ell },s)\).
The Faltings–Serre–Livné method gives an effective way to prove that two Galois representations are isomorphic up to semisimplification by showing that the trace of Frobenius agree on a finite computable set of primes. We follow the steps outlined in [13] which for practical reasons necessitates use of the prime \(\ell =2\). This prime is ramified in the acting quaternion algebra and as in Sect. 3 we can associate a representation to the 2adic Tate module.
Lemma 4.1
Proof
For \(\rho _{A,2}\) this is a direct consequence of the way that the representation has been defined. Let us now consider \(\rho _{f,2}\). The prime 31 is split in \(\mathbb {Q}(\sqrt{3})\) and the Hecke eigenvalues above these primes are distinct. Hence we can take the field by adjoining the roots of the Hecke polynomials which gives \(\mathbb {Q}(\sqrt{43},\sqrt{123})\). The completion at either of the primes above 2 in this field gives the unique unramified quadratic extension of \(\mathbb {Z}_2\) and so we can take this as the coefficient field E by [28, Corollary 1]. \(\square \)
First we must show that the residual representations are isomorphic.
Lemma 4.2
The residual representations \(\overline{\rho }_{A,2}, \overline{\rho }_{f,2}\) are isomorphic and have image \(C_3 \subset \text {GL}_2( \mathbb {F}_2).\)
Proof
Denote by \(F_A\) and \(F_f\) the fields cut out by \(\overline{\rho }_{A,2}\) and \(\overline{\rho }_{f,2}\) respectively. The field given by the 2torsion of A is isomorphic to \(A_4\) which has only one proper normal subgroup. This subgroup has order 4 and so applying the short exact sequence of Theorem 3.1, the image of \(\overline{\rho }_{A,2}\) must be \(C_3\).
To show that the representations are isomorphic let \(\psi _A\) denote the cubic character associated to \(F_A\). Extend this to an \(\mathbb {F}_3\)basis \(\{\psi _A, \chi _1 \}\) of the cubic characters of Cl\((\mathbb {Z}[\frac{1+\sqrt{3}}{2}], {\mathfrak {m}})\). We find that the prime \({\mathfrak {p}}_{37,1}\) is such that \(\psi _A({\mathfrak {p}}_{37,1}) = 0\) and \(\chi _1({\mathfrak {p}}_{37,1}) \ne 0\). So if \(\chi _f\) is the cubic character associated to \(F_f\) and \(\chi _f\) is not in the span of \(\chi _A\) then \(\psi _f({\mathfrak {p}}_{37,1})\) must be nonzero. In particular, \(\overline{\rho }_{f,2}(\text {Frob}_{\mathfrak {p}})\) must have order 3 but we find that Tr(\(\overline{\rho }_{f,2}(\text {Frob}_{{\mathfrak {p}}_{37,1}})) = \text {Tr}(\overline{\rho }_{A,2}(\text {Frob}_{{\mathfrak {p}}_{37,1}}))\) and so we can conclude that the residual representations are isomorphic. \(\square \)
Now that we have shown that the residual representations are isomorphic it remains to show that the full representations are isomorphic up to semisimplifcation. The residual images are cyclic and note that this will always be the case when the prime \(\ell \) divides the discriminant of the acting quaternion algebra. Since the images are cyclic we can use Livné’s criterion, which applies when the image is absolutely reducible.
Theorem 4.3
 1.
\(\text {Tr} (\rho _1) \equiv Tr(\rho _2) \equiv 0 \ (\text {mod } \mathscr {P})\) and \(\text {Det}(\rho _1) \equiv \text {Det}(\rho _2) \equiv 1 \ (\text {mod } \mathscr {P})\);
 2.
There is a finite set of primes T such that the characteristic polynomials of \(\rho _1\) and \(\rho _2\) are equal on the set \(\{ \text {Frob}_{\mathfrak {p}}\  \ {\mathfrak {p}}\in T\}\).
Proof
See [21, Theorem 4.3]. \(\square \)
It is now possible to show that the representations attached to A and f are isomorphic up to semisimplification.
Theorem 4.4
 1.
\(C_1: y^2= x^6 + 4ix^5 + (2i  6)x^4 + (i + 7)x^3 + (8i  9)x^2  10ix + 4i + 3\),
Bianchi newform: 2.0.4.134225.3a;
 2.
\(C_2 : y^2=x^6 + (2\sqrt{3}  10)x^5 + (10\sqrt{3} + 30)x^4 + (8\sqrt{3}  32)x^3\)
\(+ (4\sqrt{3} + 16)x^2 + (16\sqrt{3}  12)x  4\sqrt{3} + 16\),
Bianchi newform: 2.0.3.161009.1a;
 3.
\(C_3: y^2=(104\sqrt{3}  75)x^6 + (528\sqrt{3} + 456)x^4 + (500\sqrt{3} + 1044)x^3\)
\(+ (1038\sqrt{3} + 2706)x^2 + (1158\sqrt{3} + 342)x  612\sqrt{3}  1800\),
Bianchi newform: 2.0.3.167081.3a;
 4.
\(C_4 : y^2 = x^6  2\sqrt{3}x^5 + (2\sqrt{3}  3)x^4 + 1/3(2\sqrt{3} + 54)x^3\)
\(+ (20\sqrt{3} + 3)x^2 + (8\sqrt{3}  30)x + 4\sqrt{3}  11\),
Bianchi newform: 2.0.3.1123201.1b.
Proof
Restricting the representations to the cubic extension cut out by the residual representation, the mod \(\mathscr {P}\) image becomes trivial. We are then in a position to apply Livné’s criterion.
Let \(\{ \chi _1, \dots , \chi _6\}\) be a basis of quadratic characters. Any set of primes \(\{ {\mathfrak {p}}_i \}\) for which the vectors \(\{( \chi _1({\mathfrak {p}}_i), \dots , \chi _6({\mathfrak {p}}_i)) \}\) cover \(\mathbb {F}_2^6 \backslash \{0\}\) will satisfy the criterion. Following [13, §2.3 step (7)] we compute the set \(T = \{ 3, 37, 43, 61, 67, 73, 97, 103, 127, 151, 157, 193, 211, 307, 313, 343, 373, 433, 463, 499, 523, 631, 661, 823, 1321, 2197, 2557, 2917 \}.\) The traces of Frobenius agree on this set.
To complete the proof we note that this shows that the representations are isomorphic when restricting to the cubic extension cut out by the residual representations. As explained in [27, pp. 362] this means that the full representations could differ by a character. We find that the prime above 5 is inert in the cubic extension and that the traces of Frobenius agree on this prime, which forces the character to be trivial. Hence we can conclude that the two representations are isomorphic up to semisimplification.

\(T(C_1) = \{ 5, 17, 61, 73, 121, 125, 157 \}.\)

\(T(C_3)= \{ 3, 13, 19, 31, 43, 73, 79, 103, 157, 163, 181, 199, 307, 313, 397, 409, 457,\)
\(487, 643, 661, 673, 691, 823, 829, 997, 1063, 1447, 1621, 2377, 2689 \}. \)

\(T(C_4)= \{ 7, 13, 61, 79, 97 \}.\)
5 Examples
At the time of writing there are 161343 rational Bianchi newforms of weight 2 in the LMFDB [22] and these are for the quadratic fields \(\mathbb {Q}(\sqrt{d})\) with \(d=1,2,3,7,11\). Up to conjugation and twist there are only four genuine newforms for which no corresponding elliptic curve has been found. These are all accounted for by Theorem 4.4.
Curve 5.1

The surface \(A=Jac(C_1)\) has conductor \({\mathfrak {p}}_{5,1}^4 \cdot {\mathfrak {p}}_{37,2}^4\) with norm \(34225^2\).

\({\mathscr {O}}\hookrightarrow \text {End}_{\mathbb {Q}(i)}(A)\) where \({\mathscr {O}}\) is the maximal order of the rational quaternion algebra of discriminant 6.

There is a genuine Bianchi newform \(f \in S_2(\varGamma _0({\mathfrak {p}}_{5,1}^2 \cdot {\mathfrak {p}}_{37,2}^2))\) which is modular to A and is listed on the LMFDB database with label 2.0.4.134225.3a.
Curve 5.2

The surface \(A=Jac(C_2)\) has conductor \({\mathfrak {p}}_{13,1}^4 \cdot {\mathfrak {p}}_{19,1}^4\) with norm \(61009^2\).

\({\mathscr {O}}\hookrightarrow \text {End}_{\mathbb {Q}(\sqrt{3})}(A)\) where \({\mathscr {O}}\) is the maximal order of the rational quaternion algebra of discriminant 10.

There is a genuine Bianchi newform \(f \in S_2(\varGamma _0({\mathfrak {p}}_{13,1}^2 \cdot {\mathfrak {p}}_{19,1}^2))\) which is modular to A and is listed on the LMFDB database with label 2.0.3.161009.1a.
Curve 5.3

The surface \(A=Jac(C_3)\) has conductor \({\mathfrak {p}}_{7,1}^4 \cdot {\mathfrak {p}}_{37,2}^4\) with norm \(67081^2\).

\({\mathscr {O}}\hookrightarrow \text {End}_{\mathbb {Q}(\sqrt{3})}(A)\) where \({\mathscr {O}}\) is the maximal order of the rational quaternion algebra of discriminant 10.

There is a genuine Bianchi newform \(f \in S_2(\varGamma _0({\mathfrak {p}}_{7,1}^2 \cdot {\mathfrak {p}}_{37,2}^2))\) which is modular to A and is listed on the LMFDB database with label 2.0.3.167081.3a.
Curve 5.4

The surface \(A=Jac(C_4)\) has conductor \({\mathfrak {p}}_{3}^{12} \cdot {\mathfrak {p}}_{13,1}^4\) with norm \(123201^2\).

\({\mathscr {O}}\hookrightarrow \text {End}_{\mathbb {Q}(\sqrt{3})}(A)\) where \({\mathscr {O}}\) is the maximal order of the rational quaternion algebra of discriminant 6.

There is a genuine Bianchi newform \(f \in S_2(\varGamma _0({\mathfrak {p}}_{3}^6 \cdot {\mathfrak {p}}_{13,1}^2))\) which is modular to A and is listed on the LMFDB database with label 2.0.3.1123201.1b.
Notes
Acknowlegements
Many thanks to the referees for making useful suggestions and helping to improve the article. I am very grateful to both John Cremona and Aurel Page for kindly computing Bianchi newforms for me. I would like to thank Lassina Dembélé and Jeroen Sijsling for sharing their code with me and theoretical discussions which have been of great help. Finally, it is a pleasure to thank my supervisor Haluk Şengün for his suggestion of this interesting topic and great enthusiasm.
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