
// Make newform 1232.2.q.h 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_h();".
// 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_h();".
// 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 := [4, 0, 2, 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, 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_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], [1, 1, 1, 0], [0, -1, 0, -1], [0, 1, 1, -1], [1, 0, 1, 0], [-1, 0, 0, 2], [2, 0, 2, 0], [0, 1, 4, 1], [0, 1, -4, 1], [1, 0, 0, 0], [-4, 4, -4, 0], [0, 5, -2, 5], [2, 0, 0, 3], [8, 0, 0, 0], [0, 2, -2, 2], [-4, -1, -4, 0], [3, -1, 3, 0], [0, -2, -11, -2], [1, -5, 1, 0], [10, 0, 0, -3], [2, -1, 2, 0], [0, 5, -1, 5], [2, 0, 0, 6], [0, -2, -12, -2], [-13, 0, 0, -2], [-1, -6, -1, 0], [0, -7, 6, -7], [0, 11, 2, 11], [0, -6, 0, 0], [1, 0, 0, -6], [-5, 0, 0, -1], [0, 5, 12, 5], [-17, 0, -17, 0], [-8, 0, 0, 8], [0, 0, -16, 0], [7, 1, 7, 0], [-4, -8, -4, 0], [0, -7, -15, -7], [13, 0, 0, -9], [0, 2, 5, 2], [9, -7, 9, 0], [-2, 0, 0, -8], [0, -6, -2, -6], [-2, 15, -2, 0], [3, 0, 0, -10], [0, -17, 0, 0], [12, 0, 0, -5], [-4, 0, 0, -11], [-14, 10, -14, 0], [0, 0, -4, 0], [0, -13, 2, -13], [-1, 0, 0, 9], [10, -9, 10, 0], [8, 0, 0, -6], [0, 6, -11, 6], [1, 17, 1, 0], [8, -4, 8, 0], [0, -5, -7, -5], [5, -14, 5, 0], [-2, 0, 0, -7], [12, -11, 12, 0], [-16, 0, 0, 2], [-10, 0, 0, 1], [-6, 15, -6, 0], [0, -4, -15, -4], [0, -12, 4, -12], [0, -13, -13, -13], [2, 0, 0, -3], [-10, -1, -10, 0], [-4, 0, 0, 20], [14, 8, 14, 0], [0, -5, -21, -5], [4, -3, 4, 0], [0, 16, 9, 16], [-11, 0, 0, 13], [0, 7, 18, 7], [-12, 15, -12, 0], [0, -4, 30, -4], [0, 0, -25, 0], [24, -7, 24, 0], [0, 0, 0, 18], [4, 0, 0, 4], [-5, -9, -5, 0], [12, 0, 0, -18], [0, -5, -9, -5], [0, 8, 6, 8], [8, 0, 0, 0], [0, -8, 18, -8], [-13, 0, 0, 0], [-6, 0, 0, 14], [0, 24, -2, 24], [7, -5, 7, 0], [-12, 8, -12, 0], [-2, 0, 0, 2], [0, -14, 4, -14], [7, 0, 0, 9], [0, -4, 30, -4], [-12, 0, -12, 0], [0, -5, 6, -5], [0, -10, -23, -10], [14, 0, 0, 21], [16, -2, 16, 0], [24, 15, 24, 0], [0, 12, 4, 12], [6, -1, 6, 0], [1, 12, 1, 0], [5, 0, 0, -15], [0, -1, -12, -1], [28, -12, 28, 0], [-18, 0, 0, -4], [0, 20, 6, 20], [-6, 16, -6, 0], [11, 0, 0, 16], [12, 8, 12, 0], [-14, 0, 0, 13], [1, 32, 1, 0], [-37, 0, 0, 1], [-6, 5, -6, 0], [0, 5, -42, 5], [-20, 0, 0, -8], [-18, -16, -18, 0], [12, 0, 0, 3], [0, -20, -20, -20], [5, -7, 5, 0], [0, 13, -11, 13], [17, 0, 0, 18], [0, 5, 26, 5], [0, 2, -26, 2], [-6, 0, 0, 14], [0, -4, 23, -4], [-6, -30, -6, 0], [-34, 0, 0, 0], [0, 8, 0, 8], [6, 0, 0, 8], [0, 24, 6, 24], [14, 0, 0, 12], [-12, 16, -12, 0], [24, 13, 24, 0], [-2, 0, 0, 8], [8, 0, 8, 0], [-44, 0, 0, -4], [0, -2, -25, -2], [-30, -2, -30, 0], [0, 0, 0, -23], [-36, 1, -36, 0], [-18, 0, 0, -6], [-50, 0, 0, 0], [20, -20, 20, 0], [0, -1, 11, -1], [0, -13, 8, -13], [0, -20, -15, -20], [17, 0, 0, 2], [7, 0, 0, -33], [0, -9, 19, -9], [10, 16, 10, 0], [-30, 0, 0, -6], [0, 18, 20, 18], [0, 30, 1, 30], [-20, 0, 0, -1], [17, -24, 17, 0], [0, -14, -20, -14], [36, 0, 0, -13], [24, 0, 0, -4], [0, 15, -9, 15], [-20, -12, -20, 0], [-38, -6, -38, 0], [10, 7, 10, 0], [-16, 18, -16, 0], [-12, 0, 0, 0], [0, -4, 22, -4], [-10, 6, -10, 0], [-26, 0, 0, -5], [0, -11, 29, -11], [-10, -36, -10, 0], [40, 5, 40, 0], [1, 0, 0, 26], [-46, 0, 0, -3], [-26, -12, -26, 0], [28, 0, 0, 18], [0, 0, 34, 0], [0, 9, 22, 9], [30, 0, 0, -10], [24, 0, 0, -24], [0, -12, 12, -12], [50, 5, 50, 0], [-14, 29, -14, 0], [59, 0, 59, 0], [-27, 19, -27, 0], [0, 1, 0, 1], [-53, -1, -53, 0], [0, 12, -36, 12], [-22, 0, 0, 3], [0, -21, 11, -21], [0, -13, 40, -13], [-12, -28, -12, 0], [17, 8, 17, 0], [8, 24, 8, 0], [0, -25, 6, -25], [-18, 0, 0, 11], [0, 1, 47, 1], [-9, 8, -9, 0], [-3, 0, 0, -17], [0, -11, 36, -11], [-18, -21, -18, 0], [-11, 0, 0, 1], [0, -4, 0, 0], [0, -20, -28, -20], [0, -15, -4, -15], [8, 0, 0, -31], [9, -11, 9, 0], [0, -5, -2, -5], [30, 0, 0, 16], [-12, 0, 0, 8], [0, -14, 0, -14], [11, -39, 11, 0], [0, -24, -7, -24], [15, -33, 15, 0], [12, -5, 12, 0], [0, -12, 48, -12], [34, 0, 0, -20], [0, -22, 33, -22], [-9, 0, 0, 25], [0, 0, -1, 0], [0, 11, -12, 11], [58, 0, 0, -6], [15, 0, 0, 4], [0, 2, -25, 2], [-10, -29, -10, 0], [0, 21, -21, 21], [0, -10, 46, -10], [44, -20, 44, 0], [-22, -26, -22, 0], [15, 0, 0, -7], [-6, -16, -6, 0], [17, 0, 0, -19], [-21, -1, -21, 0], [0, -40, -8, -40], [0, 20, -11, 20], [19, 0, 0, 39], [-10, 0, 0, -1], [20, 26, 20, 0], [0, -28, 24, -28], [-8, 8, -8, 0], [0, 17, 34, 17], [-2, 0, 0, 19], [0, 12, -18, 12], [-52, 0, 0, 19], [-19, -25, -19, 0], [-16, 29, -16, 0], [-39, 0, 0, -5], [19, 0, 0, -8], [0, 0, -72, 0], [-24, -15, -24, 0], [20, 0, 0, 26], [-41, 16, -41, 0], [0, -18, 46, -18], [2, 44, 2, 0], [-4, 44, -4, 0], [-30, 0, 0, -11], [0, 35, -24, 35], [8, 8, 8, 0], [-12, 0, 0, -28], [-8, 14, -8, 0], [9, 0, 0, 26], [21, 8, 21, 0], [0, -23, -41, -23], [8, 0, 0, 48], [15, 0, 0, 36], [28, 0, 0, 17], [16, -38, 16, 0], [0, 20, 46, 20], [-36, -12, -36, 0], [27, -20, 27, 0], [0, 14, 8, 14], [-24, 0, 0, -11], [0, -15, 30, -15], [0, 27, 17, 27], [31, -2, 31, 0], [0, -29, 14, -29], [0, -33, 36, -33], [-41, 15, -41, 0], [2, 22, 2, 0], [17, 0, 0, 13], [7, 0, 0, -38], [0, 10, -22, 10], [0, 14, 26, 14], [13, 14, 13, 0], [26, 0, 0, 40], [-8, 25, -8, 0], [48, 0, 0, 8], [62, 0, 62, 0], [-23, 21, -23, 0], [0, -9, 36, -9], [62, 0, 0, -2], [-55, 0, 0, 18], [73, 8, 73, 0], [0, 51, 0, 51], [32, 0, 0, -28], [0, 22, -34, 22], [0, 0, 0, -15], [0, 9, -44, 9], [0, -40, 23, -40], [9, -29, 9, 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_h();".
// 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_h(: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_h();".
// 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_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1232_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 2, 5, -2, 1]>,<13,R![-7, 2, 1]>],Snew);
    return Vf;
end function;