
// Make newform 1232.2.q.g in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1232_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1232_q_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_1232_2_q_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_1232_2_q_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 := [49, 0, 7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 7, 7];
    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_1232_q();"
function MakeCharacter_1232_q()
    N := 1232;
    order := 3;
    char_gens := [463, 309, 353, 673];
    v := [3, 3, 1, 3];
    // 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_1232_q_Hecke();"
function MakeCharacter_1232_q_Hecke(Kf)
    N := 1232;
    order := 3;
    char_gens := [463, 309, 353, 673];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [1, 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+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 := [[0, 0, 0, 0], [0, 1, 0, 0], [0, -1, 1, -1], [0, -1, 0, -1], [1, 0, 1, 0], [5, 0, 0, 0], [-6, 0, -6, 0], [0, 1, 3, 1], [0, -1, 1, -1], [1, 0, 0, -2], [-4, 0, -4, 0], [0, 1, 1, 1], [3, 0, 0, -3], [4, 0, 0, 0], [0, 2, -8, 2], [1, -1, 1, 0], [2, 1, 2, 0], [0, 2, 9, 2], [-4, -3, -4, 0], [-7, 0, 0, -1], [-3, 1, -3, 0], [0, 1, 0, 1], [-8, 0, 0, -2], [0, 4, -4, 4], [-11, 0, 0, 2], [-3, 0, -3, 0], [0, 3, -5, 3], [0, 3, 3, 3], [0, 4, 0, 0], [-13, 0, 0, 2], [-8, 0, 0, 3], [0, 1, -13, 1], [-3, 6, -3, 0], [4, 0, 0, 0], [0, -2, -10, -2], [-6, -1, -6, 0], [0, -8, 0, 0], [0, -1, 2, -1], [-2, 0, 0, -5], [0, -2, 5, -2], [-12, -3, -12, 0], [-10, 0, 0, 0], [0, -2, 8, -2], [-7, -7, -7, 0], [3, 0, 0, 6], [-9, 5, -9, 0], [7, 0, 0, 3], [15, 0, 0, -1], [-8, 2, -8, 0], [0, 0, -16, 0], [0, -3, 9, -3], [4, 0, 0, -5], [1, 9, 1, 0], [-2, 0, 0, -2], [0, -4, -11, -4], [12, 3, 12, 0], [-16, 4, -16, 0], [0, -3, 10, -3], [17, 2, 17, 0], [-15, 0, 0, 3], [-5, 1, -5, 0], [12, 0, 0, 0], [-9, 0, 0, 5], [9, -3, 9, 0], [0, 4, 13, 4], [0, 0, 12, 0], [0, 1, 18, 1], [17, 0, 0, 3], [3, -9, 3, 0], [14, 0, 0, 6], [14, -8, 14, 0], [0, 3, 18, 3], [-23, -5, -23, 0], [0, 6, 5, 6], [-14, 0, 0, -3], [0, -11, 5, -11], [-3, -9, -3, 0], [0, -8, 10, -8], [0, 4, 11, 4], [-11, 3, -11, 0], [-6, 0, 0, -6], [-12, 0, 0, -8], [16, 5, 16, 0], [-16, 0, 0, 0], [0, -3, 4, -3], [0, -10, -8, -10], [-6, 0, 0, 6], [0, 8, -18, 8], [-11, 0, 0, -8], [-4, 0, 0, 10], [0, -2, -16, -2], [-18, 3, -18, 0], [-20, 4, -20, 0], [-16, 0, 0, 2], [0, 6, -2, 6], [12, 0, 0, 3], [0, -4, 10, -4], [18, 6, 18, 0], [0, 7, 3, 7], [0, 2, -3, 2], [-9, 0, 0, -7], [0, 12, 0, 0], [21, 3, 21, 0], [0, 8, -20, 8], [-37, 3, -37, 0], [-1, -10, -1, 0], [0, 0, 0, 3], [0, -3, 15, -3], [12, 12, 12, 0], [-14, 0, 0, -4], [0, 2, -16, 2], [-2, 0, -2, 0], [-11, 0, 0, -2], [24, -4, 24, 0], [-11, 0, 0, -9], [3, -6, 3, 0], [12, 0, 0, -7], [15, -9, 15, 0], [0, 1, 5, 1], [-4, 0, 0, -4], [-18, 4, -18, 0], [-7, 0, 0, -3], [0, -6, -18, -6], [-6, 3, -6, 0], [0, 5, 32, 5], [9, 0, 0, -6], [0, 9, 17, 9], [0, -6, -12, -6], [12, 0, 0, 2], [0, -10, 15, -10], [8, 6, 8, 0], [40, 0, 0, -2], [0, 0, 16, 0], [-2, 0, 0, 8], [0, 0, -6, 0], [6, 0, 0, -8], [8, 4, 8, 0], [23, 9, 23, 0], [-14, 0, 0, -8], [-40, 4, -40, 0], [4, 0, 0, -12], [0, -2, -13, -2], [-28, 6, -28, 0], [1, 0, 0, 13], [-9, 13, -9, 0], [-12, 0, 0, -6], [18, 0, 0, -8], [-36, 0, -36, 0], [0, 9, -14, 9], [0, -7, 31, -7], [0, -6, -7, -6], [-9, 0, 0, 6], [-14, 0, 0, 3], [0, -11, -16, -11], [16, 10, 16, 0], [-4, 0, 0, 2], [0, -2, 30, -2], [0, -4, 25, -4], [-7, 0, 0, -15], [-3, -18, -3, 0], [0, -6, 18, -6], [-7, 0, 0, -7], [-16, 0, 0, 16], [0, -3, 18, -3], [-10, 10, -10, 0], [-32, 2, -32, 0], [17, -15, 17, 0], [32, -4, 32, 0], [-52, 0, 0, 0], [0, 12, 6, 12], [-16, 10, -16, 0], [17, 0, 0, 3], [0, 3, -48, 3], [-34, 8, -34, 0], [45, -1, 45, 0], [7, 0, 0, -2], [11, 0, 0, 1], [-2, -16, -2, 0], [18, 0, 0, 14], [0, 0, -10, 0], [0, -5, 3, -5], [32, 0, 0, -10], [24, 0, 0, -8], [0, 0, -12, 0], [-21, 15, -21, 0], [5, -11, 5, 0], [51, -2, 51, 0], [50, 3, 50, 0], [0, 11, -3, 11], [26, -5, 26, 0], [0, -6, 14, -6], [-27, 0, 0, -9], [0, -11, 6, -11], [0, 15, -15, 15], [24, 0, 24, 0], [-15, 0, -15, 0], [-24, 4, -24, 0], [0, -5, -53, -5], [-29, 0, 0, 1], [0, -5, -10, -5], [-13, 10, -13, 0], [26, 0, 0, 7], [0, -3, 35, -3], [37, 9, 37, 0], [-16, 0, 0, 11], [-4, 16, -4, 0], [0, 0, 16, 0], [0, 13, -13, 13], [15, 0, 0, 15], [-14, 7, -14, 0], [0, -11, -23, -11], [-18, 0, 0, -12], [-32, 0, 0, 12], [0, -18, -6, -18], [-24, -3, -24, 0], [0, -4, -3, -4], [24, -7, 24, 0], [15, 9, 15, 0], [0, 4, -28, 4], [-30, 0, 0, 12], [0, 8, -21, 8], [42, 0, 0, -7], [0, -12, 3, -12], [0, 21, -5, 21], [0, 0, 0, 6], [9, 0, 0, 16], [0, 0, -45, 0], [-27, -15, -27, 0], [0, -15, 4, -15], [0, -6, -24, -6], [12, 4, 12, 0], [-24, 2, -24, 0], [40, 0, 0, -9], [10, -4, 10, 0], [-12, 0, 0, -13], [-54, 3, -54, 0], [0, 14, 30, 14], [0, -10, -23, -10], [46, 0, 0, -5], [31, 0, 0, -11], [-10, -2, -10, 0], [0, -16, -4, -16], [-36, 8, -36, 0], [0, -15, 17, -15], [11, 0, 0, 17], [0, -6, 0, -6], [37, 0, 0, -15], [10, 11, 10, 0], [-29, 13, -29, 0], [-28, 0, 0, 5], [33, 0, 0, 4], [0, 16, -28, 16], [17, 7, 17, 0], [0, 0, 0, -8], [13, -4, 13, 0], [0, 2, 16, 2], [-6, -8, -6, 0], [32, 0, 32, 0], [19, 0, 0, -11], [0, 5, -45, 5], [4, 4, 4, 0], [12, 0, 0, 24], [-60, -8, -60, 0], [19, 0, 0, 8], [7, 2, 7, 0], [0, -1, 56, -1], [-20, 0, 0, 16], [21, 0, 0, 18], [-3, 0, 0, 5], [-16, -20, -16, 0], [0, -8, 22, -8], [8, 12, 8, 0], [3, -8, 3, 0], [0, -18, -14, -18], [15, 0, 0, 7], [0, -1, 11, -1], [0, 15, -6, 15], [-29, 6, -29, 0], [0, 21, -19, 21], [0, -17, 5, -17], [18, 15, 18, 0], [-12, -22, -12, 0], [18, 0, 0, 3], [63, 0, 0, -2], [0, -10, -28, -10], [0, -6, -24, -6], [-33, -2, -33, 0], [22, 0, 0, 16], [9, -1, 9, 0], [-48, 0, 0, 0], [50, -8, 50, 0], [28, -19, 28, 0], [0, 9, 39, 9], [0, 0, 0, 6], [27, 0, 0, -8], [-37, 4, -37, 0], [0, 13, 15, 13], [12, 0, 0, 0], [0, 22, -28, 22], [13, 0, 0, -27], [0, -13, 21, -13], [0, 0, 57, 0], [-10, -15, -10, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1232_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1232_2_q_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_1232_2_q_g(:prec:=4)
    chi := MakeCharacter_1232_q();
    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_1232_2_q_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_1232_2_q_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1232_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![49, 0, 7, 0, 1]>,<13,R![-5, 1]>],Snew);
    return Vf;
end function;