
// Make newform 1232.2.q.l 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_l();".
// 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_l();".
// 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], [2, -1, 1, 0, 0, 0], [6, -18, 5, -8, 1, -1], [3, -5, 6, -2, 1, 0], [9, -21, 19, -16, 5, -2], [-9, 30, -22, 19, -5, 2]];
    Rf_basisdens := [1, 1, 3, 1, 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_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, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 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+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, 0], [0, 0, 1, 0, 0, 0], [0, -1, -1, -1, 0, 1], [-1, -1, 1, 0, 0, 1], [-1, 0, 0, 0, 1, 0], [0, -2, 0, -1, 0, 0], [2, 0, 0, 0, -2, 1], [0, 1, 0, 0, 4, -1], [0, 1, 0, 0, -1, -1], [1, 3, 0, 1, 0, 0], [7, 0, 1, 0, -7, 3], [0, 2, 2, 2, 4, -2], [6, 2, 0, -1, 0, 0], [-3, -2, 0, 3, 0, 0], [0, -1, 0, 0, 4, 1], [-2, 0, 1, 0, 2, 4], [3, 0, -3, 0, -3, 0], [0, 2, -4, -4, -2, -2], [2, 0, -2, 0, -2, -4], [4, -1, 0, 0, 0, 0], [-7, 0, -1, 0, 7, 0], [0, -5, 3, 3, 1, 5], [-2, 1, 0, -4, 0, 0], [0, 3, 3, 3, 3, -3], [10, 6, 0, -1, 0, 0], [-1, 0, -4, 0, 1, -7], [0, -5, 6, 6, 1, 5], [0, 7, 1, 1, 6, -7], [-6, 0, 3, 0, 6, 1], [-6, -5, 0, 4, 0, 0], [3, 2, 0, 1, 0, 0], [0, 5, -5, -5, 5, -5], [-5, 0, 1, 0, 5, 4], [-11, 0, 0, 0, 0, 0], [0, -6, -2, -2, 1, 6], [-4, 0, -5, 0, 4, 2], [13, 0, -2, 0, -13, -4], [0, -4, 0, 0, -9, 4], [-5, 5, 0, 3, 0, 0], [0, 6, -1, -1, 2, -6], [-1, 0, -3, 0, 1, 4], [-5, -10, 0, 1, 0, 0], [0, 2, 0, 0, 19, -2], [6, 0, -1, 0, -6, -4], [5, 5, 0, -2, 0, 0], [11, 0, -7, 0, -11, -5], [10, 8, 0, 2, 0, 0], [9, -2, 0, 0, 0, 0], [8, 0, -9, 0, -8, -5], [0, -4, -2, -2, 3, 4], [0, 2, -2, -2, -7, -2], [9, -5, 0, -5, 0, 0], [7, 0, -7, 0, -7, -2], [10, 7, 0, 2, 0, 0], [0, -7, 1, 1, -1, 7], [1, 0, -2, 0, -1, -5], [5, 0, 5, 0, -5, -1], [0, -8, -2, -2, 5, 8], [-1, 0, 3, 0, 1, -4], [-2, -1, 0, -9, 0, 0], [5, 0, -2, 0, -5, -7], [15, 12, 0, -6, 0, 0], [-6, -7, 0, 1, 0, 0], [-4, 0, 1, 0, 4, 3], [0, 1, -6, -6, -6, -1], [0, -2, 5, 5, 13, 2], [0, 4, -2, -2, 5, -4], [14, -1, 0, 8, 0, 0], [-13, 0, -5, 0, 13, -9], [-11, 2, 0, -2, 0, 0], [2, 0, 5, 0, -2, -10], [0, -5, 4, 4, 21, 5], [19, 0, 1, 0, -19, 3], [0, -10, 4, 4, -3, 10], [6, 1, 0, 0, 0, 0], [0, 4, 7, 7, -6, -4], [19, 0, 1, 0, -19, 7], [0, 2, 6, 6, 8, -2], [0, 8, 7, 7, -1, -8], [-19, 0, 5, 0, 19, 3], [7, 8, 0, 0, 0, 0], [21, -1, 0, 12, 0, 0], [-23, 0, 8, 0, 23, 0], [-2, -4, 0, 10, 0, 0], [0, 3, 2, 2, 7, -3], [0, -3, -3, -3, -18, 3], [1, -14, 0, 8, 0, 0], [0, -3, 0, 0, 16, 3], [-24, 3, 0, -6, 0, 0], [-19, 9, 0, 1, 0, 0], [0, -9, 0, 0, -21, 9], [1, 0, 9, 0, -1, -1], [3, 0, -5, 0, -3, -2], [9, 0, 0, 6, 0, 0], [0, -9, 9, 9, 11, 9], [-10, 8, 0, -5, 0, 0], [0, -1, 8, 8, 27, 1], [-7, 0, -8, 0, 7, 3], [0, 9, 6, 6, -4, -9], [0, 1, -1, -1, 5, -1], [-5, -2, 0, 4, 0, 0], [-22, 0, 3, 0, 22, -5], [2, 0, -5, 0, -2, -15], [0, 9, 0, 0, -9, -9], [-2, 0, 4, 0, 2, 9], [-12, 0, -10, 0, 12, -1], [29, 2, 0, -5, 0, 0], [0, 1, -7, -7, -17, -1], [13, 0, -5, 0, -13, 16], [-4, -9, 0, 9, 0, 0], [0, 4, -7, -7, 6, -4], [22, 0, 4, 0, -22, 2], [12, 12, 0, 0, 0, 0], [0, 0, -2, 0, 0, 4], [-7, -16, 0, 3, 0, 0], [-11, 0, 7, 0, 11, 13], [-6, 10, 0, 0, 0, 0], [-8, 0, 16, 0, 8, 10], [0, -2, 1, 1, -12, 2], [1, 5, 0, 6, 0, 0], [14, 0, 1, 0, -14, -11], [14, 10, 0, 4, 0, 0], [0, 1, 18, 18, 14, -1], [-10, 0, -1, 0, 10, 8], [0, -12, 4, 4, 12, 12], [-30, -3, 0, -6, 0, 0], [0, -13, -3, -3, -13, 13], [0, 1, 9, 9, 2, -1], [5, 1, 0, -13, 0, 0], [0, -9, 6, 6, -14, 9], [17, 0, 7, 0, -17, -4], [-5, -11, 0, 11, 0, 0], [0, 3, -19, -19, -20, -3], [14, -9, 0, 3, 0, 0], [0, 7, 2, 2, 7, -7], [-11, -18, 0, -4, 0, 0], [24, 0, -12, 0, -24, 3], [-10, 0, 3, 0, 10, -6], [-17, 5, 0, 6, 0, 0], [1, 0, -8, 0, -1, 10], [-34, 1, 0, -1, 0, 0], [0, 17, -5, -5, 8, -17], [1, 0, 5, 0, -1, -7], [12, 14, 0, 5, 0, 0], [-17, 0, 19, 0, 17, 2], [-16, -13, 0, 7, 0, 0], [-14, -22, 0, 10, 0, 0], [41, 0, 0, 0, -41, -5], [0, -12, -7, -7, 28, 12], [0, 5, -1, -1, 30, -5], [0, 20, 5, 5, 1, -20], [-18, -18, 0, 9, 0, 0], [-12, 4, 0, 3, 0, 0], [0, 7, 12, 12, 14, -7], [13, 0, 2, 0, -13, 20], [10, 19, 0, -1, 0, 0], [0, 4, 6, 6, -11, -4], [0, 13, -1, -1, -8, -13], [15, 9, 0, 7, 0, 0], [5, 0, 13, 0, -5, 0], [0, 9, -7, -7, -16, -9], [-17, 11, 0, -18, 0, 0], [-1, 4, 0, -1, 0, 0], [0, 19, 10, 10, 3, -19], [5, 0, -12, 0, -5, 13], [-15, 0, 3, 0, 15, -18], [-4, 0, 8, 0, 4, -17], [20, 0, -3, 0, -20, 8], [3, -7, 0, 3, 0, 0], [0, -9, 4, 4, 1, 9], [0, 0, -3, 0, 0, 0], [-7, 3, 0, -18, 0, 0], [0, 5, -19, -19, 0, -5], [0, 0, 18, 0, 0, -17], [13, 0, 2, 0, -13, -1], [11, -14, 0, 21, 0, 0], [7, 3, 0, -12, 0, 0], [21, 0, -5, 0, -21, 2], [-38, -2, 0, 1, 0, 0], [0, 16, 0, 0, 3, -16], [0, 10, -12, -12, -29, -10], [0, -1, 0, -19, 0, 0], [-28, -2, 0, -11, 0, 0], [0, 5, -3, -3, -38, -5], [21, 0, 4, 0, -21, -4], [22, 0, -8, 0, -22, 4], [32, 0, -1, 0, -32, 3], [32, 0, 7, 0, -32, 14], [0, -3, 9, 9, -35, 3], [54, 0, -2, 0, -54, -1], [0, -8, 0, 0, -30, 8], [1, -16, 0, 3, 0, 0], [0, 11, 10, 10, -14, -11], [0, 4, -8, -8, 30, -4], [58, 0, -1, 0, -58, 0], [-32, 0, 16, 0, 32, 4], [1, 0, 15, 0, -1, -9], [0, 3, 0, 0, 25, -3], [9, -9, 0, -3, 0, 0], [0, 13, -7, -7, 1, -13], [-27, 0, -1, 0, 27, -2], [-11, -13, 0, -4, 0, 0], [0, -7, -16, -16, -2, 7], [-47, 0, 12, 0, 47, 0], [22, -8, 0, 8, 0, 0], [18, 0, -18, 0, -18, -3], [0, -20, -7, -7, 12, 20], [0, -20, 5, 5, -29, 20], [-24, -3, 0, 15, 0, 0], [4, 0, -16, 0, -4, -7], [0, -17, -10, -10, 14, 17], [4, -6, 0, 19, 0, 0], [20, 10, 0, -13, 0, 0], [0, 9, -13, -13, -4, -9], [-4, 0, -16, 0, 4, 5], [0, 23, -1, -1, 19, -23], [6, 0, 13, 0, -6, 1], [10, 0, 16, 0, -10, 10], [0, 3, -9, -9, 3, -3], [20, 3, 0, -10, 0, 0], [0, 8, -15, -15, -31, -8], [-34, 6, 0, 1, 0, 0], [0, -27, 12, 12, -27, 27], [0, 11, 15, 15, -9, -11], [3, 6, 0, 6, 0, 0], [29, 10, 0, -2, 0, 0], [0, 12, -10, -10, -13, -12], [-10, 0, 20, 0, 10, 5], [0, 15, -7, -7, 10, -15], [0, 18, -10, -10, 8, -18], [13, 0, -21, 0, -13, 15], [34, 0, -1, 0, -34, 8], [-28, -18, 0, -8, 0, 0], [7, 0, 22, 0, -7, 22], [24, 9, 0, -7, 0, 0], [-50, 0, -2, 0, 50, 10], [0, 7, -8, -8, -2, -7], [0, -7, -20, -20, -7, 7], [-12, 3, 0, 6, 0, 0], [-4, 13, 0, -15, 0, 0], [-12, 0, -15, 0, 12, -30], [0, 5, 0, 0, 33, -5], [-19, 0, 16, 0, 19, 17], [0, 14, 10, 10, -12, -14], [0, -6, 0, 6, 0, 0], [0, 22, -1, -1, 28, -22], [6, 1, 0, 0, 0, 0], [-27, 0, -5, 0, 27, -10], [8, 0, -17, 0, -8, 26], [14, 17, 0, 10, 0, 0], [-31, -19, 0, -1, 0, 0], [0, 25, 12, 12, -22, -25], [-12, 0, -11, 0, 12, -1], [-22, 19, 0, 13, 0, 0], [-15, 0, 4, 0, 15, -1], [0, -2, 17, 17, 22, 2], [-28, 0, -14, 0, 28, -5], [-27, 0, 24, 0, 27, 2], [-2, -1, 0, 3, 0, 0], [0, 10, -1, -1, 20, -10], [-12, 0, 12, 0, 12, 8], [-13, 9, 0, -28, 0, 0], [3, 0, 13, 0, -3, 18], [-30, 16, 0, 8, 0, 0], [-9, 0, -15, 0, 9, -30], [0, -3, 7, 7, -12, 3], [-42, 12, 0, 6, 0, 0], [5, 25, 0, -24, 0, 0], [2, 27, 0, 16, 0, 0], [22, 0, 14, 0, -22, 27], [0, -16, 2, 2, 19, 16], [3, 0, -6, 0, -3, -1], [-33, 0, -2, 0, 33, -6], [0, -30, 0, 0, -4, 30], [-21, -13, 0, -6, 0, 0], [0, -6, -18, -18, 17, 6], [0, -32, 5, 5, -26, 32], [16, 0, 3, 0, -16, 12], [0, -22, 11, 11, 1, 22], [0, -31, -8, -8, 8, 31], [42, 0, 3, 0, -42, 18], [23, 0, 12, 0, -23, -15], [49, -14, 0, -4, 0, 0], [-29, -6, 0, -4, 0, 0], [0, 3, 14, 14, -2, -3], [0, -12, -3, -3, -21, 12], [24, 0, 9, 0, -24, 1], [-25, 19, 0, -18, 0, 0], [8, 0, -5, 0, -8, -13], [51, -6, 0, -9, 0, 0], [-2, 0, -5, 0, 2, -11], [-15, 0, 2, 0, 15, 13], [0, -20, -16, -16, -2, 20], [31, -9, 0, 37, 0, 0], [17, 13, 0, 1, 0, 0], [-5, 0, -7, 0, 5, 3], [0, 19, -15, -15, 31, -19], [-17, 9, 0, -11, 0, 0], [0, -11, -11, -11, -33, 11], [-14, -11, 0, -26, 0, 0], [0, 14, 1, 1, 3, -14], [0, -23, -8, -8, -3, 23], [-36, 0, -14, 0, 36, 22]];
    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_l();".
// 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_l(:prec:=6)
    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_l();".
// 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_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1232_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![9, 18, 33, 12, 7, -1, 1]>,<13,R![79, -24, -3, 1]>],Snew);
    return Vf;
end function;