
// Make newform 6480.2.a.by in Magma, downloaded from the LMFDB on 21 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_6480_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_6480_2_a_by();".
// 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_6480_2_a_by();".
// 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 := [24, 12, -11, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-6, -1, 1, 0], [0, -8, -1, 1]];
    Rf_basisdens := [1, 1, 2, 2];
    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_6480_a();"
function MakeCharacter_6480_a()
    N := 6480;
    order := 1;
    char_gens := [2431, 1621, 6401, 1297];
    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_6480_a_Hecke( : Kf := Kf)
    return MakeCharacter_6480_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, 0, 0, 0], [-1, 0, 0, 0], [0, -1, 1, 0], [2, -1, 0, 1], [0, 1, -1, 1], [0, 0, 2, -1], [1, -2, 0, 0], [3, -1, 0, 0], [2, -1, -2, 1], [-4, 1, 0, 0], [4, -2, 0, -1], [0, 0, -3, 2], [-6, 2, 2, 1], [-1, 1, 4, -1], [3, -1, 4, 1], [0, 0, 1, 0], [2, 0, 2, 2], [-4, -2, -4, 2], [8, -3, -2, -1], [0, 2, -2, 1], [-4, 0, 2, 0], [0, 2, 2, -1], [2, 1, 6, 1], [-4, 2, 4, -1], [0, 3, -4, -1], [0, 1, -3, -1], [-4, 0, -6, 0], [6, -1, -2, 0], [6, -4, 0, -1], [-4, -1, 5, 0], [4, 0, 3, 2], [4, 2, -2, 0], [-7, 0, 0, -4], [10, 2, -4, 0], [2, -1, 0, -2], [0, -3, 1, 2], [-4, 0, -4, 1], [4, -2, 4, -3], [7, -1, -4, 0], [0, 0, -3, 2], [2, 3, 0, 0], [12, 3, 4, -1], [4, 2, 0, -3], [7, 5, -4, 2], [-6, 2, 2, 2], [3, 2, 4, 2], [4, -2, 2, 2], [-10, 2, 2, 1], [8, 4, 0, 0], [4, 0, 4, -1], [12, 0, -2, 0], [4, 5, 6, -2], [10, 3, -3, -1], [14, 0, 0, 2], [9, 1, -4, 4], [12, -1, 6, -1], [6, -2, -4, 0], [4, -7, 1, 2], [-2, -5, -3, 3], [-2, -2, -2, 0], [11, 1, -4, 5], [-14, -2, 2, -3], [14, 3, -4, 3], [-2, 0, -2, 6], [13, 3, 4, -1], [-2, 5, 2, 4], [2, -2, 8, 2], [-6, 0, 8, -3], [8, 5, 2, 0], [-8, 6, -2, 1], [6, 3, 8, -1], [-4, 4, 2, 4], [4, -8, 0, 3], [11, 1, -6, -2], [9, -1, 0, 4], [-6, 6, -2, 0], [-16, -2, -4, -2], [2, -5, 9, -3], [11, -9, 2, 4], [-12, -2, -2, 0], [-10, -1, 8, -2], [32, -3, 4, 3], [0, -2, 0, -6], [2, 7, 0, -2], [-10, -6, 0, -1], [0, 0, -3, 4], [2, -2, 2, -1], [-14, -7, 10, -5], [4, -1, -5, 1], [2, -12, -4, 3], [4, 7, -10, -1], [-4, 5, 11, 1], [0, 0, 3, -4], [-1, 6, -8, 4], [-4, -4, 12, -2], [14, -8, 0, 0], [-6, -3, -5, 5], [-14, 2, -4, -6], [-6, -3, 14, -2], [6, -8, 10, 2], [9, -5, -16, 0], [-8, -4, -4, -3], [12, 2, -3, -2], [6, 1, 14, 2], [-4, -4, -16, 3], [-16, 6, 2, -1], [0, 8, -6, -2], [-4, -3, -2, -5], [-1, 0, 12, -6], [-4, 0, -14, -1], [-20, 1, -1, -3], [16, 2, 6, 2], [1, 7, 2, 4], [-14, -5, 6, -6], [-6, 7, 6, 1], [14, -2, 2, 3], [0, 6, -8, 6], [-14, 2, 0, 6], [-26, 1, -3, -5], [12, -9, -4, -4], [16, -4, -6, 4], [19, -5, 8, 2], [-14, 10, 12, 2], [17, -1, 2, -4], [-18, -3, -4, -1], [8, 6, -14, 4], [0, -1, 18, -1], [-16, -3, -9, 1], [4, -2, -4, 0], [10, -3, -6, 8], [-4, 6, 8, -7], [10, -2, 16, -4], [-20, 1, -13, 5], [-20, 0, -7, 2], [0, 3, 10, 6], [6, 2, -4, 7], [-10, -2, 0, -3], [-10, -2, 12, 2], [0, 10, 5, -2], [10, -5, -10, -4], [-14, 3, 0, -1], [0, 12, -10, 2], [-30, 4, -10, -4], [-14, 1, 2, 0], [-14, 1, -10, -5], [-4, 0, 0, -5], [10, 2, 4, 4], [38, -3, -2, 2], [29, 1, 4, 1], [4, -1, 9, -1], [-26, 3, -6, -1], [8, 6, -18, 3], [3, -7, 8, 5], [2, 12, 0, 0], [18, 5, -8, 3], [10, 5, 6, 4], [6, -7, -6, -7], [-24, -2, 4, -4], [-4, 4, -6, -4], [-18, -6, 18, -2], [-10, -4, 10, -8], [0, 6, -14, 5], [0, 0, -5, -2], [-10, 6, 4, -2], [-16, 6, -12, -1], [-10, 13, 6, 2], [0, -3, -9, 0], [25, -4, 0, -2], [-9, -9, 4, -3], [16, -6, -7, 2], [10, -13, -6, -2], [6, -1, -12, 1], [2, 6, -14, -2], [6, 0, -4, -4], [-20, -4, -3, 6], [5, -5, 2, 0], [-4, -1, 2, 7], [-8, 10, 10, 2], [-14, -3, -4, -4], [-32, 4, -2, -9], [-22, -5, 5, -1], [4, 5, 15, 2], [-18, 2, -2, -4], [13, 3, 12, 1], [18, -13, 0, 3], [4, 13, -3, 3], [-2, 14, -8, 5], [-1, 4, 4, -10], [-34, 6, 4, 2], [-14, 4, -8, -2], [-18, 10, 4, -5], [-12, -2, -16, 2], [2, -1, -12, 1], [-12, 0, -20, -7], [-22, -2, -10, 7], [5, 6, 12, -8], [44, -5, -5, 2], [4, 12, 6, -5], [-9, 5, -12, -6], [14, -13, -4, -1], [-2, 5, -22, 2], [-20, -1, 13, 2], [3, 12, -12, 4], [0, -6, 2, 8], [-5, 1, -20, 2], [-4, -7, 22, -6], [0, -6, -18, -2], [12, -10, -4, -10], [32, 0, 20, 0], [-2, 4, 0, -6], [-12, -5, 4, -7], [-8, 3, -15, 5], [-38, 4, 6, -2], [36, 7, -4, 3], [-9, 18, -4, 2], [-44, 0, 2, -4], [-26, 1, -2, 1], [-21, -9, -16, 4], [-6, 12, 12, -6], [2, 13, 4, 4], [-16, 23, 2, -6], [-4, 8, 1, 10], [-4, 5, 7, -9], [-4, -6, 12, 3], [42, -3, -10, 2], [-30, 0, -4, 0], [12, -2, -2, 4], [-4, 3, -21, 0], [-18, 15, 11, -3], [-16, -7, 3, 2], [1, 4, -12, -2], [-32, 10, 0, 4], [14, 1, -5, -1], [-32, -4, 4, 0], [-27, 9, 12, 6], [5, 0, 24, -2], [-2, 4, 20, -7], [-10, -13, 7, -1], [28, 9, 0, -1], [-46, 6, 6, 5], [35, 1, -2, 2], [40, -1, -5, 6], [32, -2, 10, -2], [-26, -2, -8, 4], [30, -9, 0, -3], [-28, 1, 1, 4], [-42, 11, 11, -1], [-7, 0, 12, 2], [-40, 0, -8, 1], [0, 2, -12, 7], [12, 4, 21, -2], [0, 8, -12, 9], [19, -14, 8, 6], [19, 1, -12, 7], [-30, 1, 9, -5], [14, -13, -4, 6], [4, 2, -4, -2], [-13, 3, -4, -1], [4, -8, 20, 3], [44, -11, -15, -2], [18, -14, 4, 9], [-14, 17, 6, 8], [-12, 7, -3, -1], [24, -12, -12, -5], [5, 0, 0, 0], [-18, 10, 16, 8], [0, -12, 3, -10], [-32, 8, 0, 4], [15, -13, -16, 0], [-24, 11, -6, 6], [-4, 12, 12, -10], [16, 2, 18, -4], [-28, -5, 8, -2], [-2, -12, 24, -2], [16, -4, 6, -3], [-18, -6, -6, 0], [8, -7, 2, -8], [31, 5, -6, 12], [18, -7, -15, -1], [-9, 1, 28, 1], [22, -14, -6, -6], [9, 9, 12, -2], [2, 0, -6, -8], [2, 0, 10, 3], [-14, 19, 10, -1], [0, 12, 6, 3], [3, -7, 16, -9], [12, 11, 0, 0], [28, 10, 8, -4], [4, 3, 2, -3], [-2, 4, -34, 2], [42, -4, 8, -1], [-6, 7, 12, 11], [-16, 5, -13, 6], [-24, -7, -10, -3], [-22, 21, -4, -6], [-30, 8, -8, -7], [0, 14, 1, 2], [-10, -12, 16, 4], [-14, -10, -10, -4], [-33, 11, -8, -3], [12, 7, 12, -4], [-12, 4, -18, -8], [-25, -2, -4, -6], [-20, 6, -10, 5], [20, 0, 14, -11], [14, -11, -4, -6], [-24, -3, -22, 1], [20, -4, 0, -10], [-12, -2, 0, -6], [-28, 5, 6, 1], [32, -6, -1, -6], [-34, 18, -2, -3], [-4, 10, -16, 17], [39, -10, 8, 4], [24, 16, 1, -4], [2, 11, -26, 1], [-20, -6, 24, -5], [-22, 3, 14, -7], [-21, 5, 10, -2], [-6, -4, -22, -7], [26, -9, -6, -1], [16, 2, -2, 9], [24, -10, 6, 4], [-35, 2, -4, 0], [22, -23, -2, 6], [26, -10, -14, 6], [-7, 15, 20, -4], [-9, 15, -4, -8], [38, 3, 8, -8], [33, -5, -12, 5], [-26, -3, 8, -10], [30, -12, -4, 5], [11, -4, 16, -4], [-18, 4, -6, 9], [-4, 5, 14, -14], [12, 10, 4, -2], [9, 10, -12, 12], [12, -25, -11, 5], [-16, 15, -1, -2], [-52, 6, 0, -4], [36, -5, 4, 9], [-18, 0, 8, 2], [3, -1, -12, -5], [-4, 14, 22, -10], [14, 3, -40, 0], [-18, 4, -4, -1], [30, -17, 0, 5], [41, -21, 0, 2], [45, 0, -24, 2], [-40, 2, 6, 0], [-10, -2, 10, 6], [16, -3, 13, -15], [-20, 1, 4, 6], [-2, -4, -16, 14], [66, -3, -6, -3], [36, -16, -19, -2], [22, 0, 4, 6], [1, -17, 4, -12], [-20, -8, 12, -6], [12, 14, -1, 2], [-15, -11, 20, -11], [-20, -2, 1, -2], [40, -16, -14, 1], [18, 16, 14, -4], [-1, 1, 16, 6], [32, -4, 14, 3], [-15, -4, -4, 0], [16, 0, -23, -2], [13, 13, -6, 4], [11, 3, 20, -6], [4, -5, 16, 1], [14, -1, -20, 6], [32, -15, 23, 10], [-12, 22, 1, -2], [0, -5, 26, 1], [26, -22, -22, 1], [-30, -19, 14, -1], [-10, -4, 10, 5], [-28, 0, 18, 2], [-34, 2, 6, 0], [28, -15, -11, -8], [-10, 8, 12, 10], [-14, -1, -2, 8], [-53, 11, -12, -7], [16, -6, 0, 6], [-40, -9, 7, -1], [-2, 2, -4, -11], [-17, -13, -4, -4], [-22, 6, 32, -4], [29, 7, 4, 2], [-16, 6, -6, 6], [-22, 12, -4, 3], [-22, -1, 8, -1], [-32, -16, -22, 9], [8, 11, -2, -4], [-6, 1, 5, 5], [15, -17, -26, 0], [36, 3, -12, 5], [-2, -7, -34, 8], [16, -2, -2, -10], [-16, 9, 29, -7], [6, 2, 6, 9], [-6, 3, 0, -15], [-14, 8, 0, 16], [4, -3, 15, 0], [42, 9, -3, -3], [4, -16, 2, 7], [14, 5, -28, 7], [-8, -6, -10, -5], [14, 8, 0, 10], [20, -2, 32, 1], [23, 14, -8, 8], [12, -20, -24, 2], [16, -2, 4, 4], [36, 5, 28, -13], [-16, 8, 6, 6], [-46, 12, -16, -1], [16, 6, 32, -8], [32, -18, 4, -6], [-36, -7, 13, 4], [31, -15, 8, 10], [60, 0, 11, -4], [-14, 2, -6, -7], [-33, 5, -12, -7], [4, 20, 8, -5], [14, 13, -23, -3], [34, -15, 10, 8], [-16, -4, 4, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6480_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6480_2_a_by();".
// 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_6480_2_a_by(:prec:=4)
    chi := MakeCharacter_6480_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_6480_2_a_by();".
// 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_6480_2_a_by( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6480_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-18, -42, -17, 2, 1]>,<11,R![-108, 144, -35, -4, 1]>,<13,R![216, 36, -39, 0, 1]>,<17,R![-72, 120, -38, -2, 1]>,<19,R![433, 0, -50, 0, 1]>,<23,R![-12, 0, 25, -10, 1]>],Snew);
    return Vf;
end function;