
// Make newform 2312.2.a.b in Magma, downloaded from the LMFDB on 14 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_2312_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_2312_2_a_b();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_2312_2_a_b();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_2312_a();"
function MakeCharacter_2312_a()
    N := 2312;
    order := 1;
    char_gens := [1735, 1157, 1737];
    v := [1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_2312_a_Hecke( : Kf := Kf)
    return MakeCharacter_2312_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0], [-2], [0], [2], [2], [-2], [0], [-4], [-6], [8], [6], [-8], [0], [12], [-8], [-6], [-4], [8], [4], [6], [-8], [2], [4], [-14], [-8], [-10], [-8], [-2], [-16], [-16], [0], [-10], [18], [14], [-10], [-8], [-18], [-14], [-18], [8], [12], [16], [24], [-8], [16], [-10], [-22], [-24], [2], [-6], [-24], [-8], [8], [12], [6], [24], [8], [-8], [0], [-10], [-30], [6], [-20], [-2], [32], [24], [-4], [32], [-6], [-14], [-14], [16], [34], [-10], [-10], [-32], [10], [-24], [24], [-22], [18], [38], [2], [6], [-26], [-4], [8], [-18], [2], [16], [12], [14], [10], [20], [-6], [-38], [30], [0], [28], [-32], [22], [-18], [36], [6], [10], [26], [-28], [14], [24], [8], [-10], [-26], [-8], [10], [-32], [8], [-18], [0], [0], [44], [-22], [8], [-8], [38], [42], [14], [-16], [-2], [0], [-30], [-36], [-34], [46], [26], [-30], [-22], [-6], [38], [2], [0], [6], [24], [26], [-30], [14], [-30], [-16], [-48], [28], [-48], [-16], [-32], [-28], [-10], [2], [42], [0], [56], [-26], [8], [6], [34], [-32], [12], [-18], [-10], [14], [0], [40], [-48], [20], [-50], [50], [34], [-32], [24], [54], [64], [16], [18], [-32], [-42], [-40], [54], [-64], [-42], [16], [-20], [-16], [38], [32], [26], [-4], [-30], [-18], [24], [32], [-40], [16], [-40], [0], [-62], [22], [-58], [36], [-62], [-64], [28], [24], [4], [40], [42], [-14], [52], [18], [16], [32], [-46], [18], [-50], [-10], [30], [14], [6], [36], [22], [40], [-46], [-40], [74], [-14], [-18], [40], [-2], [-4], [-24], [-40], [8], [46], [-16], [-30], [-28], [-64], [50], [-32], [14], [38], [-30], [-4], [8], [14], [48], [24], [40], [66], [-12], [16], [-26], [-8], [30], [22], [-4], [-8], [-64], [8], [36], [-14], [-22], [58], [-74], [16], [28], [6], [-8], [-66], [-16], [12], [-66], [2], [-60], [72], [-18], [34], [74], [-50], [-80], [8], [-16], [-8], [6], [-24], [-18], [46], [18], [48], [-32], [16], [54], [-6], [-28], [50], [-30], [74], [62], [-58], [-56], [36], [-40], [72], [22], [-78], [32], [-8], [-20], [72], [38], [60], [-74], [64], [-30], [-46], [-32], [-34], [-16], [-16], [14], [6], [18], [-18], [-24], [-80], [-24], [62], [44], [42], [-46], [-14], [0], [-72], [24], [10], [6], [-72], [-8], [14], [18], [24], [-36], [-74], [48], [4], [32], [62], [6], [10], [50], [88], [54], [-88], [-72], [-16], [24], [-8], [90], [-20], [-26], [-8], [-16], [16], [36], [-14], [50], [-42], [16], [-80], [30], [58], [-58], [-66], [-70], [72], [-90], [22], [-64], [50], [18], [0], [-22], [-10], [48], [96], [-64], [36], [92], [-88], [-16], [104], [6], [-30], [-26], [0], [-22], [-48], [0], [-26], [-18], [-62], [-38], [-44], [-66], [0], [10], [4], [-26], [26], [-48], [66], [46], [-8], [-64], [-78], [-24], [46], [-60], [-96], [94], [-90], [-72], [-84], [-46]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2312_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2312_2_a_b();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_2312_2_a_b(:prec:=1)
    chi := MakeCharacter_2312_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_2312_2_a_b();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_2312_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2312_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![0, 1]>,<7,R![-2, 1]>],Snew);
    return Vf;
end function;