
// Make newform 3330.2.h.n in Magma, downloaded from the LMFDB on 14 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3330_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3330_h_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_3330_2_h_n();".
// 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_3330_2_h_n();".
// 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 := [8, -8, 6, -6, 3, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [-2, 2, -1, 2, -1, 0], [8, -2, 4, -3, 0, -1], [4, -2, 6, -3, 2, -1], [-8, 10, -6, 3, -2, 1], [-12, 2, -6, 5, -2, 3]];
    Rf_basisdens := [1, 2, 4, 4, 4, 4];
    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_3330_h();"
function MakeCharacter_3330_h()
    N := 3330;
    order := 2;
    char_gens := [371, 667, 631];
    v := [2, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3330_h_Hecke();"
function MakeCharacter_3330_h_Hecke(Kf)
    N := 3330;
    order := 2;
    char_gens := [371, 667, 631];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [2, 0, 0, 1, 0, 0], [1, 1, 0, 2, 0, 0], [0, 0, 0, 0, 0, -2], [0, 0, -3, 0, -1, 2], [0, 0, -3, 0, 1, 1], [0, 0, 0, 0, 0, 2], [0, 0, -3, 0, 1, 2], [0, 0, 2, 0, -2, -1], [-1, 1, 2, 2, 1, -2], [-1, -3, 0, -2, 0, 0], [0, 0, 1, 0, -1, -2], [1, -1, 0, 1, 0, 0], [-1, 3, 0, -2, 0, 0], [0, 0, -5, 0, -1, 1], [0, 0, -5, 0, -1, 2], [3, 3, 0, 5, 0, 0], [4, 0, 0, 2, 0, 0], [-2, -4, 0, 0, 0, 0], [0, 0, 5, 0, -1, -1], [5, 1, 0, 5, 0, 0], [0, 0, 4, 0, -4, -4], [0, 0, -7, 0, -3, -2], [0, -4, 0, 0, 0, 0], [0, 0, -4, 0, -4, 2], [-13, -1, 0, 1, 0, 0], [0, 0, -3, 0, -3, -2], [0, 0, -3, 0, -1, 6], [-7, -5, 0, -3, 0, 0], [0, 0, -5, 0, -1, -7], [-4, -2, 0, 4, 0, 0], [7, 7, 0, 2, 0, 0], [-14, -2, 0, 0, 0, 0], [-4, 0, 0, 2, 0, 0], [-3, -7, 0, -6, 0, 0], [0, 0, 9, 0, 3, 0], [0, 0, 8, 0, -8, -6], [-7, 1, 0, 2, 0, 0], [0, 0, 3, 0, -5, 3], [8, -4, 0, -4, 0, 0], [0, 0, -6, 0, -2, 1], [0, 0, -6, 0, 0, 8], [-6, 2, 0, 4, 0, 0], [0, 0, 11, 0, 5, -3], [9, 9, 0, 8, 0, 0], [8, 6, 0, 1, 0, 0], [0, 0, -15, 0, -1, -2], [-14, 4, 0, 0, 0, 0], [0, 6, 0, 8, 0, 0], [0, 0, 14, 0, -2, 1], [0, 0, -12, 0, 0, -4], [0, 0, -5, 0, 3, 1], [0, 0, 6, 0, -6, 4], [10, 0, 0, -1, 0, 0], [4, 0, 0, -4, 0, 0], [-10, -6, 0, -2, 0, 0], [0, 0, 6, 0, -10, -10], [0, 0, 10, 0, 6, 0], [0, 0, -6, 0, -2, 6], [-3, 5, 0, 14, 0, 0], [11, -5, 0, -5, 0, 0], [0, 0, 8, 0, 4, -3], [0, 0, -2, 0, 8, 8], [5, 1, 0, -2, 0, 0], [0, 0, -5, 0, -1, 7], [0, 2, 0, 12, 0, 0], [0, 0, -12, 0, 8, 8], [-14, -2, 0, -4, 0, 0], [0, 0, -5, 0, -9, -14], [-12, 4, 0, -4, 0, 0], [12, -6, 0, 11, 0, 0], [-2, 2, 0, 0, 0, 0], [2, -2, 0, -10, 0, 0], [0, 0, 16, 0, 8, -10], [0, 0, -3, 0, 5, -2], [18, 2, 0, 8, 0, 0], [0, 0, -6, 0, -2, 0], [0, 0, 4, 0, -4, 8], [-10, -6, 0, -10, 0, 0], [0, 0, -6, 0, 4, -4], [0, 0, 12, 0, 4, 13], [14, 0, 0, -8, 0, 0], [0, 0, 24, 0, 4, -5], [3, -1, 0, 11, 0, 0], [0, 0, 10, 0, 10, -8], [0, 0, -3, 0, -7, -18], [0, 0, 21, 0, -7, -6], [0, 0, -2, 0, 2, -2], [0, 0, -7, 0, -5, 4], [0, 0, 5, 0, 11, 7], [0, 0, -6, 0, -10, -10], [14, 2, 0, -6, 0, 0], [0, 0, -13, 0, 11, 7], [0, 0, -12, 0, 4, 16], [0, -6, 0, -4, 0, 0], [-1, -13, 0, -16, 0, 0], [0, 0, 30, 0, -6, -2], [0, 0, -2, 0, 0, -12], [0, 0, -3, 0, 7, -8], [0, 0, 2, 0, -2, 2], [0, 0, -1, 0, -11, -8], [0, 0, 12, 0, -12, -16], [3, 3, 0, 6, 0, 0], [0, 0, -6, 0, -10, -8], [0, 0, 1, 0, 7, -2], [-36, 2, 0, 0, 0, 0], [-12, 8, 0, -14, 0, 0], [19, 3, 0, -8, 0, 0], [0, 0, -2, 0, -14, -14], [-15, 1, 0, 2, 0, 0], [4, -6, 0, 12, 0, 0], [-3, -3, 0, -24, 0, 0], [0, 0, 4, 0, 12, -3], [-21, 3, 0, 0, 0, 0], [0, 0, -9, 0, 1, 14], [0, 0, -14, 0, 14, 4], [0, 0, 2, 0, 0, 0], [0, -4, 0, -4, 0, 0], [0, 0, 13, 0, 1, 6], [2, -8, 0, 8, 0, 0], [14, -10, 0, 4, 0, 0], [0, 0, 5, 0, 3, -2], [-21, 11, 0, 0, 0, 0], [0, 0, -6, 0, 0, -8], [0, 0, 9, 0, 13, 10], [-6, -2, 0, -10, 0, 0], [0, 0, 16, 0, 0, -6], [-11, -7, 0, -2, 0, 0], [7, 7, 0, -6, 0, 0], [0, -6, 0, -19, 0, 0], [10, 2, 0, -16, 0, 0], [0, 0, 8, 0, 2, 12], [19, -7, 0, 14, 0, 0], [0, 0, 0, 0, -12, -4], [-11, -19, 0, -18, 0, 0], [-31, 5, 0, 15, 0, 0], [0, 0, 10, 0, -14, -14], [0, 0, -2, 0, -2, -12], [-12, 0, 0, -4, 0, 0], [-28, -6, 0, -12, 0, 0], [-19, -1, 0, 13, 0, 0], [0, 0, 13, 0, -5, -6], [0, 0, 3, 0, 7, -14], [-18, -18, 0, -4, 0, 0], [0, 0, -6, 0, -8, -16], [0, 0, -29, 0, 15, 10], [0, 0, 15, 0, 7, 19], [-2, 12, 0, -1, 0, 0], [25, -11, 0, 18, 0, 0], [9, 13, 0, 4, 0, 0], [0, 0, -15, 0, 7, 14], [-6, 0, 0, -25, 0, 0], [0, 0, -20, 0, 24, 16], [0, 0, -19, 0, -5, -9], [0, 0, -23, 0, 15, 3], [-15, 3, 0, -22, 0, 0], [0, 2, 0, 8, 0, 0], [-12, -8, 0, -16, 0, 0], [0, 0, -9, 0, -3, 0], [30, 8, 0, 4, 0, 0], [0, 0, 10, 0, 6, 12], [-33, -9, 0, -4, 0, 0], [0, 0, 27, 0, 7, 2], [20, -10, 0, -17, 0, 0], [0, 0, -10, 0, -6, -7], [0, 0, -20, 0, 14, 4], [-13, -17, 0, -20, 0, 0], [0, 0, 8, 0, -22, -4], [0, 0, 19, 0, 11, 9], [0, 0, -39, 0, 9, 2], [0, 0, 2, 0, -14, 7], [10, -8, 0, -20, 0, 0], [-6, 18, 0, 4, 0, 0], [0, 0, 30, 0, -10, -8], [0, 0, 33, 0, -3, 9], [2, -2, 0, 16, 0, 0], [2, 4, 0, 7, 0, 0], [6, 0, 0, -8, 0, 0], [0, 0, 26, 0, -10, -4], [0, 0, 19, 0, 3, 21], [0, 0, 10, 0, -2, -10], [0, 0, -7, 0, -7, -10], [6, 16, 0, 1, 0, 0], [12, -14, 0, -4, 0, 0], [-17, -9, 0, 2, 0, 0], [0, 0, 41, 0, -1, -10], [0, 0, 6, 0, -14, -4], [-2, 2, 0, 18, 0, 0], [0, 0, -14, 0, -2, 0], [15, 3, 0, 13, 0, 0], [0, 0, 15, 0, -9, -7], [8, 6, 0, -12, 0, 0], [-21, -25, 0, -19, 0, 0], [-26, 8, 0, 0, 0, 0], [0, 0, 10, 0, 10, -12], [0, 0, -26, 0, 10, -6], [8, -6, 0, 4, 0, 0], [0, 0, 14, 0, 10, 14], [0, 0, -2, 0, 16, 24], [8, 4, 0, 10, 0, 0], [-39, -7, 0, -10, 0, 0], [17, 13, 0, 20, 0, 0], [-25, 7, 0, 0, 0, 0], [0, 0, -32, 0, -8, -10], [-16, -4, 0, -14, 0, 0], [-37, -1, 0, -1, 0, 0], [0, 0, 36, 0, 8, -12], [21, 21, 0, 16, 0, 0], [0, 0, -41, 0, 5, -2], [0, 0, -1, 0, -5, -22], [0, 0, 42, 0, 6, 0], [-13, 15, 0, 15, 0, 0], [0, 0, 10, 0, 14, -15], [-23, -3, 0, 4, 0, 0], [0, 0, 2, 0, -10, 16], [0, 0, -6, 0, 10, 32], [0, 0, 0, 0, -8, -12], [11, -5, 0, -14, 0, 0], [-2, -6, 0, 12, 0, 0], [-22, 2, 0, 4, 0, 0], [-11, 1, 0, -20, 0, 0], [0, 0, -16, 0, -8, 16], [-31, 13, 0, -5, 0, 0], [0, 0, -14, 0, -8, -8], [44, 2, 0, 12, 0, 0], [38, 2, 0, 10, 0, 0], [48, 6, 0, 9, 0, 0], [0, 0, 27, 0, -17, -3], [21, 17, 0, 6, 0, 0], [-8, 4, 0, -28, 0, 0], [24, 8, 0, 20, 0, 0], [-18, 4, 0, 0, 0, 0], [11, 19, 0, -3, 0, 0], [34, -8, 0, 1, 0, 0], [-18, 4, 0, 0, 0, 0], [0, 0, -24, 0, -4, 26], [0, 0, -3, 0, 5, 7], [0, 0, -27, 0, -13, -17], [0, 0, -7, 0, -13, 0], [0, 0, -13, 0, -21, 5], [-9, -3, 0, -29, 0, 0], [0, 0, 41, 0, 1, -6], [4, 10, 0, 16, 0, 0], [0, 0, -36, 0, 0, 3], [0, 0, -16, 0, 16, 12], [0, 0, 5, 0, 13, 27], [-59, -3, 0, -2, 0, 0], [0, 0, 16, 0, -8, -18], [0, 0, -38, 0, -6, 14], [-16, -4, 0, -10, 0, 0], [0, -10, 0, -11, 0, 0], [0, 0, 36, 0, 16, -12], [0, 0, -24, 0, 6, 12], [13, -19, 0, 0, 0, 0], [-26, -4, 0, -16, 0, 0], [-15, -3, 0, -11, 0, 0], [-17, -13, 0, 10, 0, 0], [0, 0, -31, 0, 17, 14], [0, 0, 40, 0, 8, 10], [0, 0, 24, 0, -20, -12], [-8, 6, 0, -4, 0, 0], [-13, 3, 0, 6, 0, 0], [0, 0, 14, 0, -6, 8], [25, 9, 0, 4, 0, 0], [-44, -8, 0, 0, 0, 0], [0, 0, 6, 0, 14, 24], [11, 7, 0, -1, 0, 0], [0, 0, -16, 0, -16, 2], [0, 0, 22, 0, -24, -4], [0, 0, 73, 0, 3, -4], [0, 0, -14, 0, 14, 24], [0, 0, 8, 0, -4, 1], [-22, 8, 0, 12, 0, 0], [-30, 16, 0, -5, 0, 0], [19, 23, 0, -1, 0, 0], [0, 0, -46, 0, 4, 12], [3, 1, 0, 22, 0, 0], [0, 0, -1, 0, -1, -17], [-29, -7, 0, 15, 0, 0], [0, 0, -30, 0, -2, -19], [56, -6, 0, 3, 0, 0], [-12, -10, 0, -12, 0, 0], [0, 0, -18, 0, -6, 14], [-24, 8, 0, 32, 0, 0], [0, 0, -11, 0, -17, -10], [-7, -11, 0, 2, 0, 0], [0, 0, -31, 0, -1, 19], [0, 0, 30, 0, -6, -4], [0, 0, 15, 0, 7, 10], [0, 0, 64, 0, -4, 4], [-10, -8, 0, -16, 0, 0], [-33, 15, 0, -6, 0, 0], [-35, -3, 0, -10, 0, 0], [-22, 4, 0, -8, 0, 0], [-10, 10, 0, -6, 0, 0], [6, 2, 0, 4, 0, 0], [0, 0, 7, 0, -5, 5], [-15, 1, 0, -27, 0, 0], [0, 0, 3, 0, 11, 34], [62, 14, 0, 8, 0, 0], [36, 0, 0, 26, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3330_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3330_2_h_n();".
// 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_3330_2_h_n(:prec:=6)
    chi := MakeCharacter_3330_h();
    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_3330_2_h_n();".
// 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_3330_2_h_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3330_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 4, -5, 1]>,<11,R![-16, -16, -1, 1]>,<13,R![256, 0, 320, 0, 36, 0, 1]>,<41,R![-172, -56, 1, 1]>],Snew);
    return Vf;
end function;