
// Make newform 945.2.bj.g in Magma, downloaded from the LMFDB on 21 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_945_bj();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_945_bj_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_945_2_bj_g();".
// 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_945_2_bj_g();".
// 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 := [3, -12, 19, -15, 10, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [6, 0, 1, 5, -1, 1], [0, -3, 2, -5, 1, -1], [6, -21, 19, -16, 5, -2], [-9, 33, -22, 19, -5, 2]];
    Rf_basisdens := [1, 1, 3, 3, 3, 3];
    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_945_bj();"
function MakeCharacter_945_bj()
    N := 945;
    order := 6;
    char_gens := [596, 757, 136];
    v := [3, 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_945_bj_Hecke();"
function MakeCharacter_945_bj_Hecke(Kf)
    N := 945;
    order := 6;
    char_gens := [596, 757, 136];
    char_values := [[-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 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+1)]];
    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 := [[-1, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, -2, -1, 0, -1, 0], [0, 1, 1, -1, 0, 1], [1, 2, 1, 1, 2, -2], [1, -2, 0, -2, 1, 0], [4, -1, -2, -2, 2, 1], [1, 0, 0, -1, 1, 0], [-1, -5, -3, -3, -1, 6], [-1, 2, 1, -2, 1, 1], [1, 2, 0, 1, -5, -4], [5, -1, 0, 2, -1, 0], [0, 0, 4, 4, 0, 0], [1, -2, 0, 1, -8, 0], [4, -2, 1, 2, -4, 1], [3, 3, 6, 3, 3, -6], [-9, -1, -2, -5, -3, 1], [-10, -5, -3, -5, -10, 3], [6, -4, 2, 2, 4, -4], [0, -1, -2, 1, 0, -2], [-5, 7, 0, -5, 0, 3], [8, 2, 0, -4, 2, 0], [-4, -1, 0, -4, 5, 9], [-2, 8, 1, 1, 2, -2], [3, 0, 3, 0, 3, -3], [-3, 4, 8, 5, 0, -4], [-4, 5, 10, 6, 0, -5], [2, 1, 5, 1, 2, -5], [-2, 0, 1, 1, -6, -2], [4, -1, 3, 5, -1, 0], [5, -4, 0, 5, -10, -6], [-5, 7, 4, -7, 5, 4], [-2, 2, 4, 4, -10, -8], [-1, 0, 0, 5, -3, 0], [-3, -1, 2, -1, -3, -2], [-1, 0, -3, 0, 1, -3], [3, -3, 0, 3, -1, -3], [-7, -4, 0, 8, -4, 0], [-1, 5, 0, -1, -4, -3], [1, -8, -2, 8, -1, -2], [-7, 5, -5, -5, 1, 10], [-1, -7, -14, 3, -9, 7], [2, 1, 8, 1, 2, -8], [-1, 3, 4, 4, -7, -8], [-1, -5, -1, 5, 1, -1], [8, -1, 1, 3, -1, 0], [9, 7, 2, 2, 21, -4], [8, -4, -3, -4, 8, 3], [-1, -2, -4, -7, 1, 2], [-21, -2, -4, -11, -7, 2], [-9, 3, 1, 1, -17, -2], [14, -3, 0, 3, -14, 0], [22, 1, 3, 1, 1, 0], [-2, -5, 0, -2, 7, 9], [9, -14, -2, 14, -9, -2], [9, 0, 3, 0, 9, -3], [-5, -2, -4, -7, -1, 2], [1, 3, -6, 3, 1, 6], [3, 11, 2, 2, 13, -4], [0, -2, -1, 2, 0, -1], [-7, 2, 3, -1, 2, 0], [0, 6, 0, 0, 6, 0], [22, 1, -3, 1, 22, 3], [21, 3, 6, 3, 12, -3], [-21, 4, 8, 7, -10, -4], [2, 6, 0, 2, 14, -10], [2, -1, 4, 6, -1, 0], [-14, 9, 2, -9, 14, 2], [-5, 1, -1, -1, -7, 2], [-6, -6, -12, -6, -6, 12], [-9, -1, -2, -3, -4, 1], [5, -13, 1, 13, -5, 1], [10, -11, 0, 10, -9, -9], [21, -6, 4, 16, -6, 0], [-13, 20, 0, -13, 8, 6], [5, 0, -4, 0, -5, -4], [0, -6, -12, -6, -3, 6], [15, 5, 10, 9, 8, -5], [-9, 14, 7, -14, 9, 7], [9, 3, 6, 0, 3, 0], [-14, 5, -3, -13, 5, 0], [4, -8, -5, 8, -4, -5], [-13, 5, -5, -5, -11, 10], [5, -6, -12, -3, -2, 6], [-4, -1, -2, 0, -3, 1], [-3, 1, 0, 0, -5, 0], [3, 4, 0, 3, 0, -10], [-10, 5, -3, -13, 5, 0], [-7, 9, 18, 0, 9, 0], [1, -17, 0, 1, -14, 15], [8, 11, 6, 11, 8, -6], [-14, 9, -1, 9, -14, 1], [-19, 1, -6, -6, -25, 12], [-7, 6, 0, -7, 11, 8], [14, -1, 12, 14, -1, 0], [-3, 3, 0, -3, -21, 3], [2, -13, -15, -13, 2, 15], [-8, -5, -10, -4, -7, 5], [2, 11, 0, 2, -14, -15], [-8, -1, -12, -10, -1, 0], [-13, 19, 1, -19, 13, 1], [-10, -10, -9, -10, -10, 9], [-12, -1, -2, 6, -10, 1], [22, 3, 15, 3, 22, -15], [-1, -7, -8, 7, 1, -8], [18, -3, 0, 6, -3, 0], [-6, 21, 0, -6, 33, -9], [1, 5, -6, -5, -1, -6], [11, -5, -1, -1, 19, 2], [14, 0, 0, 18, -2, 0], [-1, -14, -12, -14, -1, 12], [-16, 1, -6, -6, -19, 12], [7, -11, 2, 11, -7, 2], [-10, 0, -6, -6, 0, 0], [-17, 17, 3, -17, 17, 3], [-3, -7, -8, -8, 3, 16], [-4, 14, 21, 14, -4, -21], [-13, -7, -14, -27, 0, 7], [6, -3, 4, 4, 1, -8], [12, -5, 2, 5, -12, 2], [-4, 8, -8, -24, 8, 0], [-3, 21, 0, -3, 24, -15], [-17, 6, 14, -6, 17, 14], [0, -1, -2, 22, -12, 1], [-3, 0, 1, 1, -8, -2], [-13, 21, 0, -13, 17, 5], [16, -20, 0, 16, -17, -12], [0, 1, 5, 5, -9, -10], [-12, 5, 10, 10, -6, -5], [2, 1, 9, 1, 2, -9], [-18, 9, 1, 1, -29, -2], [14, -12, 0, 14, 8, -16], [26, -4, -2, 6, -4, 0], [0, 3, 0, 0, -9, -3], [4, -11, 2, 2, -7, -4], [3, -6, -15, -6, 3, 15], [13, -22, -11, 22, -13, -11], [0, -9, -3, 15, -9, 0], [-11, -3, 11, 3, 11, 11], [5, -15, 0, 0, -5, 0], [23, -1, -2, 21, 0, 1], [-29, -9, -15, -9, -29, 15], [11, 3, 1, 1, 23, -2], [23, -27, 6, 27, -23, 6], [12, -9, 3, 21, -9, 0], [5, -13, 1, 1, -5, -2], [-24, 3, 0, 3, -24, 0], [9, 9, 18, -3, 15, -9], [13, 2, 4, 21, -2, -2], [-15, 12, 0, -15, 23, 18], [12, -3, -12, -6, -3, 0], [-11, 7, -2, -16, 7, 0], [4, 7, 0, 4, 10, -15], [-22, 7, 8, 7, -22, -8], [1, 17, 8, 8, 3, -16], [13, -10, 0, 13, -17, -16], [-9, 24, 0, -9, 39, -6], [10, 19, 11, 11, 17, -22], [24, 9, 6, 9, 24, -6], [-16, 5, 10, -6, 0, -5], [-25, -11, -6, -6, -49, 12], [23, 3, -9, -15, 3, 0], [14, -13, 0, 14, -4, -15], [8, 6, -7, -6, -8, -7], [-26, 4, 12, 4, -26, -12], [13, 18, 26, 18, 13, -26], [-8, -3, 0, 3, 8, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_945_bj_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_945_2_bj_g();".
// 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_945_2_bj_g(:prec:=6)
    chi := MakeCharacter_945_bj();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_945_2_bj_g();".
// 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_945_2_bj_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_945_bj();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, 6, 1, -6, 1, 3, 1]>,<11,R![147, -42, -59, 18, 25, -9, 1]>],Snew);
    return Vf;
end function;