
// Make newform 1776.2.bz.e in Magma, downloaded from the LMFDB on 29 April 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1776_bz();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1776_bz_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1776_2_bz_e();".
// 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_1776_2_bz_e();".
// 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 := [1, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_1776_bz();"
function MakeCharacter_1776_bz()
    N := 1776;
    order := 6;
    char_gens := [223, 1333, 593, 1297];
    v := [6, 6, 6, 5];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1776_bz_Hecke();"
function MakeCharacter_1776_bz_Hecke( : Kf := false)
    N := 1776;
    order := 6;
    char_gens := [223, 1333, 593, 1297];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [1, 0, -1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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], [1, 0, -1, 0], [0, 1, 0, 0], [-3, 1, 3, -2], [-3, -2, 0, 1], [2, 2, 2, 0], [-2, -6, 1, 6], [1, 5, 1, 0], [2, 0, -4, 2], [2, 0, -4, 3], [-3, 0, 6, 1], [0, -7, 0, 4], [-5, -2, 5, 4], [1, 0, -2, 7], [7, -2, 0, 1], [0, 4, 2, 4], [8, 0, -4, 0], [-2, -5, -2, 0], [5, -5, -5, 10], [-7, 1, 7, -2], [2, 12, 0, -6], [-3, -1, -3, 0], [0, 2, -10, 2], [10, 0, -5, 0], [-5, 0, 10, -10], [0, -14, 0, 7], [3, 0, -6, -1], [-11, -1, 11, 2], [0, 1, 0, -1], [0, -2, 0, 2], [0, 8, -4, 8], [4, -2, -2, 2], [-3, 20, 0, -10], [0, 4, 0, 4], [-10, 10, 0, -5], [2, 2, -2, -4], [0, 5, 16, 5], [10, 9, -5, -9], [-2, -2, -2, 0], [0, -11, -6, -11], [-12, 0, 24, 0], [-8, 3, 8, -6], [4, 0, -8, 12], [1, 0, -2, 16], [0, 1, 14, 1], [-1, 0, 2, 7], [5, -18, 0, 9], [-4, 8, 0, -4], [-3, -19, -3, 0], [26, 1, -26, -2], [3, -4, 0, 2], [22, -11, -11, 11], [4, -6, 4, 0], [11, 0, -22, 9], [10, 0, -5, 0], [0, 12, 0, -24], [16, -8, 0, 4], [0, -11, -7, -11], [-10, 1, -10, 0], [22, -6, -11, 6], [-11, -11, -11, 0], [-6, -13, 6, 26], [-15, -18, 0, 9], [2, 19, -1, -19], [-2, 12, 1, -12], [0, -7, -4, -7], [-12, 22, 6, -22], [3, -8, -3, 16], [-2, 0, 4, 6], [0, 1, -6, 1], [-6, 14, 3, -14], [8, 0, 0, 0], [-14, 6, 14, -12], [20, 7, -20, -14], [0, -3, 15, -3], [-7, -1, -7, 0], [-12, -13, -12, 0], [-20, 14, 0, -7], [-4, 0, 8, -6], [-30, -12, 15, 12], [0, 11, -11, 11], [-6, 0, 12, -17], [7, 9, 7, 0], [23, 8, 0, -4], [0, 4, 0, 0], [14, -12, 0, 6], [0, 14, 0, 0], [5, -10, 5, 0], [4, 2, -2, -2], [-5, -7, -5, 0], [4, 0, -8, 4], [4, -30, -2, 30], [-12, 0, 24, -20], [15, -30, 0, 15], [-3, 9, -3, 0], [10, 27, -5, -27], [0, -7, -16, -7], [6, -2, -6, 4], [20, 8, 20, 0], [2, 0, -4, -21], [17, 0, -34, -1], [4, -1, -2, 1], [-10, 0, 20, 14], [-11, 0, 22, 0], [0, -8, 0, -8], [-16, 14, 8, -14], [-14, 22, -14, 0], [-3, 16, 0, -8], [-17, 7, 17, -14], [0, 14, -7, 14], [46, 9, -23, -9], [0, -5, 22, -5], [0, 8, -6, 8], [-1, 34, 0, -17], [-36, -6, 18, 6], [0, 2, 5, 2], [-1, 0, 2, 33], [-15, 3, -15, 0], [4, 9, 4, 0], [-6, -14, 6, 28], [0, 1, 0, 0], [-14, 10, 14, -20], [26, -22, 0, 11], [34, 17, -17, -17], [0, 7, 17, 7], [-20, -18, 10, 18], [0, 0, 0, 46], [0, -2, 2, -2], [26, 6, 26, 0], [14, 0, -14, 0], [8, -16, 0, 8], [13, 21, -13, -42], [-3, 2, 0, -1], [-12, 37, 6, -37], [0, 10, -15, 10], [4, 0, -8, 34], [-6, 3, 6, -6], [10, -12, 0, 6], [-4, 26, 2, -26], [-16, 10, -16, 0], [4, 20, -4, -40], [0, -12, 0, 24], [0, 6, -22, 6], [12, -20, 12, 0], [28, 10, -14, -10], [0, 14, -14, 14], [-16, -25, 8, 25], [-21, 0, 42, -20], [-15, 0, 30, 19], [0, 8, 12, 8], [-26, -6, 0, 3], [3, -24, -3, 48], [4, -16, 4, 0], [-22, 28, 0, -14], [9, 17, 9, 0], [6, 0, -12, 18], [4, 0, -8, -12], [-21, 4, 21, -8], [0, 12, 27, 12], [0, 7, -40, 7], [44, 18, -22, -18], [0, -2, 10, -2], [22, 2, 22, 0], [0, 24, 0, 24], [40, 2, -20, -2], [0, 13, 29, 13], [-5, 0, 10, -5], [12, -14, -6, 14], [-1, -48, 0, 24], [-6, 0, 12, 17], [-18, 3, 9, -3], [12, 21, -6, -21], [-22, -2, -22, 0], [33, -4, -33, 8], [-17, -21, 17, 42], [16, -24, 16, 0], [-4, -34, 2, 34], [0, 21, -4, 21], [8, -16, 0, 8], [-4, 3, 4, -6], [-18, 0, 36, -30], [24, 0, 24, 0], [-12, -27, 6, 27], [-19, 0, -19, 0], [-7, 33, 7, -66], [6, 14, 0, -7], [16, 17, -16, -34], [12, -24, 12, 0], [11, 2, 11, 0], [-15, -23, 15, 46], [13, 0, -26, -34], [0, 4, 12, 4], [-16, -24, -16, 0], [-20, -3, 20, 6], [-9, -23, 9, 46], [0, -28, -15, -28], [-18, 20, 9, -20], [-6, 0, 12, 34], [18, -6, -18, 12], [4, 42, -2, -42], [-26, 0, 52, -2], [10, 28, 0, -14], [0, -17, 14, -17], [0, -4, 43, -4], [6, 20, 0, -10], [-18, 5, -18, 0], [0, -12, -16, -12], [0, -4, 16, -4], [17, 0, -34, 6], [0, 16, 0, -32], [-24, -22, 12, 22], [-2, 0, 4, 25], [-9, 0, 18, -41], [0, 7, 45, 7], [38, 2, 38, 0], [-38, -16, 0, 8], [-2, 50, -2, 0], [-7, 0, 14, 18], [0, 32, 0, -32], [56, 4, -56, -8], [0, 8, 20, 8], [33, -3, -33, 6], [-5, -8, 5, 16], [-14, -17, 7, 17], [0, -7, 7, -7], [0, 0, 0, -7], [38, -36, 0, 18], [9, 17, -9, -34], [42, -6, -42, 12], [-1, 0, 2, 9], [-6, 22, 0, -11], [0, 10, -22, 10], [0, -15, 37, -15], [27, -4, 0, 2], [9, -1, -9, 2], [43, 7, -43, -14], [0, 10, -22, 10], [-4, 61, -4, 0], [-22, -18, -22, 0], [-13, 0, 26, -21], [-6, 0, 12, 18], [18, 0, -36, -38], [-35, 10, 0, -5], [8, -35, 8, 0], [45, 4, 0, -2], [18, -22, 18, 0], [-8, -20, -8, 0], [-8, -44, 4, 44], [0, -20, -8, -20], [-23, 0, 46, -27], [-28, 0, 56, 3], [35, -12, 0, 6], [0, -15, 9, -15], [4, -68, 4, 0], [20, -2, -10, 2], [0, 25, 15, 25], [22, 4, -22, -8], [13, -18, 0, 9], [0, 1, -40, 1], [1, 0, -2, 20], [-14, -1, 7, 1], [-2, -49, 1, 49], [2, 8, -2, -16], [0, -12, -30, -12], [-20, 6, -20, 0], [-41, -15, 41, 30], [48, 1, -48, -2], [1, 2, 1, 0], [0, 0, 4, 0], [8, 0, -16, -18], [-36, 46, 18, -46], [-6, 0, 12, -10], [-3, 0, 6, 28], [-70, 7, 35, -7], [5, -24, 0, 12], [-53, 7, 53, -14], [55, 26, 0, -13], [-2, 14, -2, 0], [0, -20, -25, -20], [-26, 23, 13, -23], [45, 26, 0, -13], [-22, 14, -22, 0], [68, 8, -68, -16], [-38, 8, 0, -4], [38, 7, -19, -7], [0, 15, 19, 15], [-35, 0, 70, -16], [-12, -32, 0, 16], [-4, 0, 8, -32], [14, 30, -7, -30], [-18, 0, 36, -18], [6, -53, -3, 53], [-21, -44, 0, 22], [17, -1, -17, 2], [0, -9, -44, -9], [0, 19, -6, 19], [-14, -76, 0, 38], [0, 37, -14, 37], [37, 9, 37, 0], [8, -32, 0, 16], [1, -28, 1, 0], [6, -8, 0, 4], [68, -8, 0, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1776_bz_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1776_2_bz_e();".
// 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_1776_2_bz_e(:prec:=4)
    chi := MakeCharacter_1776_bz();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1776_2_bz_e();".
// 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_1776_2_bz_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1776_bz();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 0, -1, 0, 1]>,<11,R![6, 6, 1]>],Snew);
    return Vf;
end function;