
// Make newform 9464.2.a.z in Magma, downloaded from the LMFDB on 01 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_9464_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_9464_2_a_z();".
// 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_9464_2_a_z();".
// 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
        poly := [32, 6, -12, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [2, -6, -1, 1], [-6, -1, 1, 0]];
    Rf_basisdens := [1, 1, 2, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_9464_a();"
function MakeCharacter_9464_a()
    N := 9464;
    order := 1;
    char_gens := [2367, 4733, 2705, 9297];
    v := [1, 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_9464_a_Hecke( : Kf := Kf)
    return MakeCharacter_9464_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, 0, 0, 0], [0, 1, 0, 0], [-1, 0, -1, 0], [1, 0, 0, 0], [0, 1, 0, 2], [0, 0, 0, 0], [4, 0, 0, -1], [3, 0, 1, 2], [1, 1, -1, 0], [1, -2, -1, -1], [1, -1, 1, -1], [-2, 3, 0, 2], [-6, 1, -2, -1], [3, 0, -1, 1], [1, 3, -3, 1], [-1, -2, 1, -1], [6, -2, -2, -2], [-2, 3, -2, 3], [-2, 3, 0, -1], [2, 2, -4, 1], [-1, -1, 1, -1], [3, -1, 1, -2], [-5, 2, -3, 0], [-1, -2, -1, -4], [-5, -3, 1, 1], [-6, 3, -4, -2], [4, 2, 0, -2], [2, 0, -2, 2], [0, -2, 0, 1], [1, 3, 1, 0], [2, 1, -4, 1], [-4, 4, 0, 6], [6, 2, 2, 3], [-2, -2, 0, 1], [8, -5, 2, 0], [-2, 2, 0, -3], [-4, 5, 2, 2], [-4, -4, 4, -2], [7, 2, 1, -4], [-8, 2, 0, 1], [-1, -2, -1, -5], [-10, -1, 0, 2], [2, 0, 2, -2], [6, 2, 0, 4], [2, 9, -2, 1], [8, -2, 2, -1], [1, -2, 5, -1], [-5, -3, 3, -3], [8, 4, 4, -2], [4, 2, 2, 0], [9, -7, 1, 0], [-8, -2, 0, 0], [1, 2, 1, -6], [-6, -3, -2, 0], [10, -4, 2, -1], [-3, 0, -3, -5], [-12, 7, 0, 3], [4, -5, -2, -5], [3, 0, 5, 3], [-12, 0, 0, 5], [4, -1, 2, 1], [5, 6, 1, 4], [-13, -2, -3, -4], [-10, 0, -4, -1], [2, 2, -2, 5], [0, -7, 4, -5], [-6, -1, -8, -3], [1, 3, 1, 0], [0, -4, 4, 4], [-5, 2, -5, 4], [-16, 1, 0, -5], [-2, 2, 4, 3], [2, -6, 4, 1], [-4, 2, 2, 4], [4, -6, 6, -7], [20, -1, 2, 5], [-4, 2, -2, 10], [-5, -8, 3, -8], [4, -2, 2, 4], [-5, 2, 3, 0], [-2, -3, -2, 2], [-14, 5, -4, 6], [14, -10, 2, -4], [8, 0, -2, -6], [2, -2, 2, 2], [1, 4, 1, 5], [-16, 0, -2, 8], [6, -2, 10, -1], [-6, 10, -4, -4], [16, -10, -2, -3], [14, 2, -4, 3], [9, -2, 7, -4], [-18, 6, 6, 6], [-22, 0, 2, 4], [-14, 3, 2, 0], [-14, 4, 2, 2], [-15, -2, -7, 2], [-4, -4, 2, 2], [-6, 11, -4, 1], [14, -6, 6, 3], [-5, 0, -1, -7], [12, -3, 8, -5], [8, 3, 2, -1], [25, -1, 1, 0], [9, -14, 1, -3], [6, 4, 0, 4], [-5, 8, -11, 4], [-9, 2, -9, 0], [11, -1, 1, 6], [-14, -4, 0, 4], [22, 0, 4, -9], [-8, 7, 0, 1], [18, -4, -4, -4], [24, -6, 4, 2], [22, 2, 2, 8], [11, -11, -5, -10], [4, -4, -8, -6], [-6, -6, -4, 1], [-2, 6, -4, -2], [19, -10, 7, 1], [5, -8, 5, 2], [17, 1, -3, -6], [6, -9, 0, -14], [32, 3, 2, 1], [-23, 2, -5, -6], [-13, 2, 1, 5], [28, -3, 6, 8], [-2, -4, 4, 7], [6, -2, -6, -8], [-15, -8, 1, 2], [-8, -10, 6, -5], [0, -16, 0, 0], [3, 9, 5, 8], [-9, -8, 1, -1], [-17, 3, 5, 1], [23, 7, 1, 3], [-18, 8, -4, 2], [-7, -6, -5, 2], [2, -7, 6, -7], [15, 12, -3, 3], [20, -4, 4, -12], [-2, -8, 0, -2], [18, -11, 0, -1], [-4, -12, 4, -10], [2, 2, -6, -5], [-12, -9, 2, -15], [-1, -8, 3, -6], [10, 0, -4, -2], [14, -3, 4, -9], [4, -4, 4, 4], [-26, -3, 0, -6], [14, -12, 0, -6], [18, -6, 6, 2], [4, 6, 4, 4], [33, 2, -3, -1], [1, -4, 9, 3], [8, 11, -6, 7], [-5, 11, -3, 3], [-32, 8, 0, 1], [13, 2, -3, -2], [14, 0, -2, 0], [3, -4, -7, -5], [8, 4, 10, 3], [10, -11, -4, -15], [16, 10, -4, 5], [-5, 6, 1, 18], [10, 3, 0, -1], [-12, 3, -16, -5], [0, 0, -4, -3], [-28, 2, -8, -5], [-20, -6, -6, -1], [6, 0, 4, 2], [-13, -7, -3, 4], [-18, 4, -10, 5], [-8, -14, 0, -8], [20, -6, 6, 8], [28, -11, 2, -5], [-18, 8, -8, -2], [10, 6, 4, 9], [-10, 7, 4, 8], [-20, 14, -4, 8], [-2, -1, -6, -4], [32, 0, 2, -2], [-31, -7, -1, -9], [-12, 2, -6, -7], [2, 10, -6, 3], [23, 2, -3, 1], [-11, 2, 7, -2], [-6, -14, 4, -12], [-5, -1, -1, -5], [16, -4, -12, -5], [-30, 7, 4, 7], [9, -6, 5, -5], [11, -6, 15, 2], [-1, -14, -1, -21], [-4, 18, -4, 13], [32, -8, 12, 5], [-5, -8, 1, -7], [-7, -10, 9, -2], [-10, 12, -6, -4], [-2, -15, -4, -4], [12, 6, 2, -7], [-9, 0, 7, 0], [-24, 0, 6, 2], [5, 4, 3, -4], [-12, -4, -2, 12], [-7, 6, 3, 6], [-37, -2, 3, -1], [-10, 26, -6, -1], [-2, -5, 0, 9], [-11, -13, 1, 6], [-16, 9, 2, 0], [-16, 11, 2, 3], [-4, -4, 8, 6], [-5, 1, 3, -7], [-5, 11, 1, 21], [-9, 1, 1, 12], [16, -18, -2, -18], [-28, 6, -4, -4], [14, -11, 4, -12], [-23, -4, -9, 11], [22, 5, -8, 1], [40, -2, 2, 2], [-10, -10, -6, 4], [-14, 9, -12, -3], [7, 10, 5, 5], [-24, 2, -6, -4], [-15, 3, -13, -14], [-10, 2, -12, -5], [12, 7, 0, -12], [36, -2, 6, -10], [24, 2, 14, 1], [20, 0, 12, 16], [8, -15, 0, -11], [-42, 1, 0, -7], [-27, 7, -15, 1], [-10, 19, -2, 13], [-16, 1, -4, 3], [7, 4, -9, -1], [18, -16, 6, 8], [-4, 13, -10, -7], [-22, 21, -4, 1], [-48, 8, 0, 4], [-2, 4, 0, -6], [45, -9, -5, -9], [-48, 0, -2, -3], [-31, -5, -3, -3], [-27, 16, 3, 8], [10, -7, 10, 8], [-5, 11, 5, 16], [36, -1, -4, -3], [-42, 4, -4, 6], [14, 6, -10, 2], [14, 0, 8, 2], [18, -7, -2, -7], [36, -8, -4, -6], [9, 4, -9, 3], [28, 2, 8, -14], [-8, 3, -4, -5], [-19, -3, 3, 3], [25, -8, -7, -1], [5, 22, -3, 5], [59, -2, 3, 0], [-38, -11, 4, -10], [-10, -19, 6, -7], [42, -8, 10, 6], [6, 7, 10, 13], [-13, -5, 13, 7], [-4, -5, 8, 4], [-4, 6, 10, 4], [26, -15, 0, -20], [10, -9, -2, -6], [2, 5, 18, 5], [-25, 8, -1, 13], [36, 6, 0, 11], [12, -8, -4, 0], [2, -6, 0, -3], [-14, -21, 12, -14], [-8, -6, -10, -4], [-24, -12, 0, 8], [-6, -14, -6, -18], [-14, 2, 6, 16], [16, -2, 12, -14], [7, -2, -13, 2], [-6, 0, -10, -8], [19, 9, 1, 10], [25, 7, 5, -14], [4, 0, -6, -2], [0, -13, -2, -19], [-24, 0, -12, -17], [-5, -20, 9, -15], [16, -5, -10, -3], [18, 0, 12, -4], [19, 12, 5, -4], [5, 12, -5, 3], [-12, -8, 18, -2], [-10, 12, 2, 12], [-12, -11, 0, -17], [-12, -1, 0, -6], [-19, 8, -9, 6], [2, 2, 10, -1], [-26, 3, 0, 18], [-23, -12, 5, 5], [17, -4, 13, 7], [-13, -20, 11, -5], [0, 24, -8, -1], [14, 2, 14, 0], [-8, 6, -16, -3], [-14, 14, -18, 8], [35, -6, 1, 5], [4, 16, -6, 5], [16, 14, -4, 7], [40, -5, 12, -5], [-15, 6, -11, 11], [22, -8, 8, 9], [32, 2, 10, -16], [5, -22, 11, -8], [52, -4, 6, 7], [-25, 2, -9, 8], [11, -13, 3, -6], [0, -3, -22, -1], [-24, 6, 10, 10], [10, 5, -8, 12], [-16, 2, -8, -20], [-4, -8, -2, -18], [8, 2, 20, 11], [32, 3, -2, 11], [-29, -14, -3, 0], [38, -12, 2, 0], [-20, 8, 10, 4], [-40, -11, 10, 0], [-9, -2, -3, 5], [-42, -4, 8, 1], [5, -18, 15, -18], [36, 3, 0, 12], [7, -4, 1, 4], [10, -18, 14, -3], [6, -21, 14, -15], [-16, 12, 2, 18], [-18, -10, -10, -6], [-26, -10, 6, 3], [8, -17, 0, -3], [-4, -8, -2, -26], [2, 15, 0, 15], [2, 9, 4, 2], [6, -8, -2, 8], [14, -10, -10, -9], [6, 2, 4, 1], [-1, -11, 7, -15], [-18, 13, 0, 2], [-21, 0, 1, 18], [8, -8, -4, 5], [26, -11, 12, -18], [-38, -2, -2, -8], [32, 0, 2, 18], [18, 0, 22, 13], [19, 1, -5, 1], [-5, 4, -15, -6], [20, 13, -16, 3], [-16, 22, 6, 17], [-6, 8, -4, 18], [-6, 12, -4, 18], [11, 1, -3, 0], [-6, -23, 6, -10], [34, 1, 10, -2], [-11, 5, 5, 0], [11, -4, -1, -6], [-40, 4, 2, -1], [-1, -13, -9, 2], [-22, 6, -14, 0], [26, -5, 22, 4], [14, -4, 12, -9], [5, -6, -1, 5], [-4, -2, -14, -7], [-15, -10, -5, 5], [12, -24, 8, 2], [18, 7, 12, 17], [-8, 9, -8, 19], [0, 2, -18, -10], [42, -6, 20, 18], [-29, -12, -9, -6], [38, -6, 16, 24], [-38, -2, -8, -17], [-42, 20, -2, 1], [25, 6, 3, 18], [27, -6, 13, 0], [-30, 0, -16, -13], [16, -15, 4, -13], [-2, 6, -8, 7], [-8, -18, -2, -3], [-14, 16, -8, -6], [-59, -18, 9, -8], [-24, -3, 6, -5], [-28, -8, 8, -6], [-4, 26, 0, 10], [9, 5, -11, -2], [4, -9, 14, -1], [26, -22, 4, 2], [17, -18, 17, -3], [8, 16, -6, 0], [-43, 12, -3, 2], [-1, -8, 1, 5], [12, 2, 10, -7], [41, -11, 11, 9], [37, 8, 9, 4], [28, -18, -2, -1], [28, -11, -4, -15], [6, -12, -6, 3], [57, -2, -5, -2], [69, 2, 3, 4], [4, 8, 2, 30], [-3, 6, 11, -1], [55, -10, 11, 1], [-14, -16, -2, 6], [-17, 1, -17, -2], [-32, -10, -4, -1], [20, 14, 4, 4], [-8, -12, -4, 14], [-58, 5, -18, -11], [-32, 8, -2, 17], [-25, -4, 9, -9], [16, 8, -6, -14], [6, 14, 16, 1], [-36, -2, -10, -25], [10, -4, 4, 8], [33, -12, -15, -14], [-33, -2, -25, -5], [-8, 18, 10, 10], [21, -8, 11, -4], [22, -6, 10, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9464_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9464_2_a_z();".
// 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_9464_2_a_z(:prec:=4)
    chi := MakeCharacter_9464_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_9464_2_a_z();".
// 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_9464_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9464_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![32, 6, -12, -1, 1]>,<5,R![56, -18, -15, 2, 1]>,<11,R![488, 22, -44, -1, 1]>,<17,R![32, -140, 82, -16, 1]>],Snew);
    return Vf;
end function;