
// Make newform 3234.2.a.bk in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3234_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_3234_2_a_bk();".
// 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_3234_2_a_bk();".
// 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 := [14, 0, -8, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [0, -4, 0, 1], [-4, 0, 1, 0]];
    Rf_basisdens := [1, 1, 1, 1];
    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_3234_a();"
function MakeCharacter_3234_a()
    N := 3234;
    order := 1;
    char_gens := [1079, 199, 2059];
    v := [1, 1, 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;

function MakeCharacter_3234_a_Hecke(Kf)
    return MakeCharacter_3234_a();
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, 0], [1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 0], [-1, 0, 0, 0], [0, -1, 0, 1], [0, 0, 1, -2], [0, 1, -1, 1], [-2, -1, 2, -2], [0, 0, 0, 2], [4, 0, -1, -1], [2, 1, 0, -4], [0, 0, -1, 2], [2, -1, 2, 2], [4, 1, -1, -1], [2, 1, 0, 4], [8, 0, 0, 2], [4, -1, 0, -1], [4, -1, 4, 2], [0, 2, 0, 2], [0, 2, -1, -4], [4, 1, 0, -6], [0, 0, 1, 3], [4, -3, 0, -1], [-4, 2, 0, -1], [4, 2, -2, 3], [8, -2, 3, 3], [2, -3, 0, 0], [-2, 2, 0, 4], [-4, 0, -2, -2], [4, -1, -4, -6], [4, 0, -1, -3], [2, -2, 0, -2], [0, -1, 3, 1], [-4, -2, -4, 4], [4, 2, -4, 0], [0, 2, 1, -4], [8, 0, 0, -4], [4, -4, 0, -2], [-4, 0, -4, 1], [-8, -1, -4, 6], [16, 2, -3, 6], [-6, 3, -2, -6], [2, 0, -4, 4], [-14, 2, 0, 2], [8, 0, -1, 3], [2, 1, 2, 2], [8, -4, -1, -1], [0, -2, 5, 3], [0, 0, -1, 2], [-6, 0, 0, -4], [-12, 2, 0, 0], [0, -2, 3, -8], [4, 2, -2, -6], [4, -1, -4, -9], [-4, -3, 4, 6], [16, 0, -3, -4], [12, 2, 0, 6], [8, -2, -6, -8], [-10, -2, 0, -8], [-8, -1, 3, -11], [0, 0, -4, 5], [-12, 3, -3, -1], [12, 3, -5, 11], [12, -2, 2, 7], [14, -1, 4, 4], [-8, 1, 0, 6], [2, 1, 4, 0], [-6, 5, -4, 0], [-4, -1, 6, 7], [0, -1, -4, 5], [-8, -5, 4, -6], [12, -2, 5, 1], [-6, -2, 0, -10], [-4, -4, 4, 8], [4, 1, 5, -11], [14, 2, -8, 0], [-8, 0, -3, -12], [-14, 2, 0, 10], [24, 2, 3, -10], [8, 2, 2, -4], [2, 4, -4, -4], [4, 1, -4, -6], [0, -2, 4, -3], [-4, 0, 0, -2], [0, 5, -4, 6], [4, -4, 2, 10], [-2, 2, 0, 16], [-8, 0, 2, 5], [8, 4, -4, -4], [12, -2, -8, -8], [4, 6, 2, 0], [-10, 5, -8, -4], [-20, 0, 0, -8], [8, -3, 0, -18], [-20, -4, -4, 6], [0, 0, -3, -20], [-4, 5, -4, -7], [0, -3, -3, 11], [6, -2, 0, 14], [20, 0, -4, -2], [6, 6, 0, -10], [4, -2, 3, 25], [6, 8, -4, 4], [-20, 4, 0, 2], [-4, 2, 2, -3], [-20, 4, 2, 10], [-24, 0, -1, -2], [0, -4, 4, -18], [-8, 2, 1, 14], [8, 2, 0, 16], [16, 2, 2, -6], [-16, 0, -6, 0], [0, 0, -2, -4], [2, -3, -4, 8], [0, -8, 2, -12], [0, 4, 4, -6], [-4, 3, 3, 15], [-2, -2, 0, -16], [14, -3, 4, 12], [-16, -4, 9, -2], [6, -3, 4, 12], [-4, -6, 12, -5], [-12, -1, 0, 10], [-12, 0, -6, 6], [-6, 0, 8, 0], [-10, 1, 4, 12], [-16, -1, -5, -1], [16, 0, -1, 11], [-32, 1, 4, 7], [4, -4, 8, 6], [-12, 2, -4, -12], [-10, -5, -4, 20], [-14, -3, 0, 12], [0, 2, -5, -8], [-24, -2, -7, 6], [-32, -4, -1, 6], [-16, 7, -7, -5], [8, -6, 1, 2], [-2, 7, 4, -4], [-28, -1, -5, 5], [4, 2, 4, 16], [-18, 3, -8, 4], [-12, 4, 4, 0], [-8, 0, -1, 26], [8, -3, 5, 1], [-16, 7, -12, -13], [-16, 8, -7, 0], [32, -2, 4, 10], [14, -1, -2, 14], [-22, 0, 8, 4], [8, -1, -6, -15], [4, 0, 0, -20], [0, 2, 0, -12], [-4, 5, 4, -6], [6, 3, -6, -2], [16, 6, 0, 0], [-8, 3, 4, 15], [16, -4, 1, -8], [-12, -2, 12, -9], [-4, 8, -12, -4], [26, -3, 4, -4], [-16, -10, 4, -4], [16, 4, -10, 4], [4, 4, -10, -16], [-4, -1, -3, -19], [0, 6, -12, 0], [4, -9, 6, 11], [-6, 0, 0, 0], [0, 6, -5, 4], [22, -3, 0, 12], [8, -8, 5, 0], [-12, 0, -8, 12], [-22, 3, 0, 20], [28, -2, 3, 3], [-16, -3, 2, 15], [12, -8, 0, -10], [-6, 0, -4, -28], [-16, -4, 10, -22], [12, 3, 2, -3], [0, 2, -4, 20], [4, 8, 5, -5], [-22, -5, 0, 12], [-8, 6, -13, 0], [8, -6, -4, 6], [0, 0, 11, -4], [6, -6, 8, -2], [4, 8, -8, -12], [-10, -5, 0, 8], [-8, 6, -6, -2], [-12, 4, -2, -11], [2, 7, -8, -4], [-12, 5, -4, 18], [-16, 0, 3, 26], [4, -2, 0, 32], [8, 1, -4, -11], [2, 9, 0, -8], [10, 3, 0, 8], [16, 6, -3, -14], [16, -8, 6, 10], [4, -2, 4, 4], [-4, -4, 8, 2], [8, 4, 7, -8], [8, 0, -9, -4], [28, 6, -12, 12], [8, 2, -1, 12], [32, -6, 5, 9], [-4, 2, -4, -20], [30, 3, 8, -12], [12, 2, 0, -8], [-10, -1, 0, -28], [8, 12, 1, -2], [16, -4, -12, -8], [-8, -2, 16, -6], [12, -2, 4, 6], [24, -4, -2, -23], [-20, -1, -8, 10], [-16, -10, 5, -12], [-8, -2, -4, 34], [-6, -4, 0, 4], [-12, 4, -10, 18], [28, 6, -4, 18], [-8, -2, 2, 14], [24, -2, -4, -24], [-12, -6, 9, 3], [-10, 4, -16, 4], [-20, -9, -6, 7], [16, 4, -4, -6], [56, 0, -2, 2], [42, -3, -8, -12], [-2, -3, -8, 24], [0, 3, 11, 5], [24, -8, -4, -16], [22, 1, 8, 0], [-24, 0, -10, 0], [36, -4, 8, -2], [-8, 2, 0, -1], [-12, -2, 0, -12], [8, -2, -8, -4], [16, 1, -9, 15], [12, 7, -4, 6], [28, -7, 1, 7], [-8, -6, 21, -3], [-26, 2, 8, 4], [-16, 2, -9, -12], [24, 2, -10, -6], [36, 2, 1, -3], [-32, 8, -4, -6], [12, 8, -12, 18], [0, 3, -8, -18], [6, -2, -8, -10], [8, 2, 7, 18], [-16, 4, -12, 10], [24, 6, 4, -9], [-16, 2, 0, 15], [14, 1, 4, -32], [2, -9, -4, 12], [4, -1, 3, -15], [8, 0, -13, -16], [8, -4, 9, -12], [8, -4, 16, 24], [4, 0, 4, -16], [0, -5, 0, -13], [-44, -1, -2, -1], [8, -5, -2, 17], [4, 6, -2, -26], [10, 5, -4, 4], [36, -7, -6, -15], [20, 8, -4, -6], [-20, 4, 8, 26], [24, -10, 9, 4], [-32, -4, 4, -8], [-24, 8, 2, 15], [-16, 7, -4, -14], [8, 0, 11, 16], [32, 4, 11, -25], [20, -2, 0, 8], [-2, -6, 16, -10], [10, -10, 8, 0], [-28, 6, -9, -15], [-12, 2, -2, 4], [16, -2, 8, -8], [-12, -2, -6, -20], [0, 3, 6, 35], [4, 5, -1, 5], [-2, 3, 6, 26], [4, -2, 4, -6], [8, 0, 8, -8], [4, -2, 7, -13], [16, 10, -3, -20], [10, 7, -4, -24], [8, -8, -10, -4], [-14, -2, 0, -44], [12, -2, -3, -9], [24, -2, 14, 16], [32, 2, -6, 21], [20, 6, -3, 17], [-56, 0, 3, 4], [-4, -2, -1, -7], [-12, -1, -13, -7], [-8, -2, 3, -10], [-22, -4, 8, 4], [64, 1, -4, 10], [-40, -5, 12, -14], [-28, 2, 4, 4], [-30, 8, 0, -2], [16, -4, 12, -16], [40, -10, 5, 10], [-16, 10, -8, -14], [40, -10, 2, 4], [20, 12, -10, 0], [22, -2, 4, 16], [26, -6, 8, 4], [-16, 13, 4, 18], [-28, -8, 4, 4], [8, -2, 3, 28], [-4, -6, -2, 26], [0, 7, -8, 10], [4, -2, 10, 25], [2, 12, -8, 0], [4, -1, 11, -31], [-32, -4, 4, 0], [0, 2, 4, -35], [-14, 7, -16, 0], [-26, -15, 12, -12], [-8, 4, -6, -51], [40, -6, 4, -16], [-36, 4, 0, 12], [40, -1, 4, -2], [6, 4, -24, 12], [-10, 10, -16, -18], [-6, 14, -12, -8], [4, 0, -4, -14], [-4, 0, -1, 41], [-28, -2, 0, -6], [-20, 2, 0, -24], [-10, -1, -4, 36], [-16, -8, -5, -12], [4, 6, -32, 5], [36, -2, -10, 0], [-18, 10, -20, -4], [8, 8, -10, 16], [-4, 16, -12, 7], [-18, -1, 8, -36], [-14, -6, 12, -20], [4, 4, -8, 32], [-8, 12, -3, 22], [-8, -8, 0, -28], [8, -10, 4, -24], [8, -6, 9, -10], [-4, 1, 21, -5], [40, 2, 4, 2], [18, 1, -12, 0], [-32, 2, -8, 8], [-28, -6, -10, -4], [8, -2, 11, -12], [12, -3, 9, -3], [-28, 0, 1, -1], [-18, 5, -16, 4], [8, 14, 0, 2], [-36, -14, 10, -12], [-8, -10, -5, 24], [24, -14, 12, 2], [12, 2, -12, 8], [56, 2, -6, 8], [-36, 2, -8, -12], [24, -2, 8, -25], [64, 1, 0, 10], [-6, 4, -16, -12], [-60, 0, -4, 0], [-16, -6, 6, -4], [8, 5, -8, -2], [2, 14, -16, 10], [-8, 4, -16, -16], [26, -6, 8, 16], [-20, -4, 4, -12], [-36, 3, 4, -14], [8, -14, -3, -18], [8, 8, -15, 22], [-16, 4, -1, 44], [32, -10, 3, -2], [-10, -6, 24, -16], [8, -9, 0, 34], [-30, -1, 8, -28], [-52, -3, -5, 13], [36, -7, -7, -15], [-6, -1, 8, -32], [-8, -6, -17, 2], [46, 9, 14, -30], [72, 1, -3, 5], [-12, -18, -4, 8], [-64, -4, -11, 14], [4, -6, 8, -16], [12, 4, -18, 14], [16, -8, 0, 26], [22, 11, 12, -12], [0, 4, -12, 4], [24, 9, -6, 21], [-24, 3, -8, -2], [-2, -4, -8, -20], [8, -5, -4, -9], [30, -4, -16, -24], [4, -10, 12, 24], [16, -1, 2, 39], [0, -4, 16, -27], [-52, 4, 0, -10], [-22, 6, -16, -8], [22, 13, 0, 8], [-36, -2, 0, -8], [4, -4, 8, 8], [-16, -12, 15, -32], [24, -4, -12, -34], [4, -6, 32, 8], [4, -8, 0, 22], [30, -5, 4, -12], [0, -4, 19, 10], [-36, -7, 16, 2], [-20, -4, -19, -3], [12, -3, 6, 7], [16, 0, -4, -12], [30, 10, 4, -20], [32, 3, -4, -21], [16, -8, 16, 10], [-16, -6, -1, -11], [-64, -2, -2, 11], [0, 8, 1, -6], [-12, 3, 0, -30], [2, 0, 12, 0], [-44, -10, 4, -12], [-4, 2, -14, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3234_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3234_2_a_bk();".
// 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_3234_2_a_bk(:prec:=4)
    chi := MakeCharacter_3234_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_3234_2_a_bk();".
// 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_3234_2_a_bk( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3234_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![56, 0, -16, 0, 1]>,<13,R![164, -32, -36, 0, 1]>,<17,R![-8, 32, -32, 0, 1]>],Snew);
    return Vf;
end function;