
// Make newform 3234.2.a.bl in Magma, downloaded from the LMFDB on 19 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_bl();".
// 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_bl();".
// 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 := [2, -4, -6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -4, -1, 1], [3, 2, -1, 0], [-3, 4, 2, -1]];
    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], [-1, 0, 0, -1], [0, 0, 0, 0], [1, 0, 0, 0], [-2, 0, -1, 1], [-1, 0, 2, 1], [-3, 0, -1, 0], [-2, -1, 1, 1], [-2, -2, 0, 2], [-1, 1, 0, 3], [2, 3, -1, -1], [-3, -4, 2, -1], [-2, -3, -1, -1], [-1, 2, 1, 0], [-2, 1, -3, -3], [0, 4, -2, -2], [-6, 0, -3, -1], [-2, 3, 1, -1], [2, 0, 2, -4], [-1, -4, 0, -1], [-2, -3, 3, 1], [-1, -5, 0, -1], [-6, 4, -1, 1], [0, 5, 0, 4], [-6, 1, 2, 0], [-1, 1, 4, 3], [-2, 5, -3, -3], [-2, 6, 2, -2], [-4, -4, -2, 2], [-2, -5, 1, -1], [-11, 3, -2, -1], [0, 4, 2, -4], [1, 2, 1, -2], [-8, -2, 2, -2], [-4, 6, -2, -2], [9, -2, 2, 3], [0, 0, -4, -4], [4, -4, -6, -6], [-8, 5, 0, -4], [-2, 5, 3, 5], [5, 12, 2, -1], [-14, 3, 1, 1], [-2, 12, 0, 4], [-8, -8, 2, 0], [3, -5, -2, -7], [-2, 3, -3, 5], [11, -1, -2, 1], [11, -7, -2, 1], [-1, -14, 0, -1], [-2, -8, -4, 4], [0, -10, -2, 2], [-9, -4, -4, 3], [2, -4, 6, 4], [-6, -2, 1, -5], [-6, -1, -3, -1], [-1, -4, 0, -1], [8, 2, 0, -4], [-2, 4, 0, 6], [-10, -2, -2, 2], [9, -2, 1, -2], [12, 3, 6, 2], [-13, 10, 3, 6], [11, 0, 3, -2], [2, -3, -8, -2], [-6, 3, 7, 7], [-2, -7, -5, -3], [14, -1, -1, 3], [2, -3, -3, -7], [0, 8, -3, -3], [-2, -6, -1, 1], [6, 1, 3, -3], [5, 1, 2, -5], [0, 0, -6, 0], [4, 12, 4, 4], [21, 0, -3, 2], [-14, -6, 2, 2], [5, -8, 0, -3], [4, -12, -2, 4], [-17, 2, 0, 7], [10, -8, 4, 6], [6, 0, 4, 8], [18, -1, 1, 3], [-16, -3, 2, -6], [-12, -8, 2, 2], [-10, -5, 9, 3], [-12, 0, 6, 2], [10, -2, -2, -6], [-18, -5, 6, -4], [0, 8, 4, -4], [16, 10, -2, 2], [22, 4, -4, 6], [14, -5, -1, -1], [-4, -8, 0, -8], [2, 21, -1, 5], [8, -8, -2, 10], [7, 0, 4, -5], [-2, 10, -1, -3], [15, 10, 7, -6], [4, 0, 2, 0], [10, -6, 4, -6], [0, -8, -2, 0], [1, 5, 4, 9], [-6, 16, -4, -4], [-6, -14, 0, -2], [-6, -17, 0, -2], [-14, -6, 0, -2], [-11, 12, -2, -5], [18, -2, 4, 6], [-19, -10, -4, 1], [4, 6, -6, -2], [4, -2, 0, 0], [-2, 2, 6, -4], [14, -2, -2, -4], [-6, -1, -1, 7], [-14, 6, -2, 0], [4, -4, 2, 6], [19, -8, 7, 10], [-22, 6, -2, -10], [-6, -15, -3, -3], [1, 6, 0, 1], [-6, 7, -5, -1], [12, 1, 2, 10], [-14, -3, 7, -7], [14, 2, -8, -6], [-2, -16, 8, 0], [-2, -17, -5, 3], [7, 2, 1, 4], [3, -1, 2, -11], [-10, 4, -11, -9], [-10, -10, 0, 2], [0, 10, 6, 2], [-14, 5, 1, -3], [22, 7, -5, -1], [-27, -12, -8, 5], [-15, 18, 0, 9], [-11, -14, -4, 1], [11, -12, -7, 0], [7, 0, 10, 1], [-10, 7, -5, -1], [-3, 20, 7, 4], [4, 22, -2, 6], [-2, -19, 1, 1], [8, 12, -4, 8], [-13, -6, 0, 11], [9, 6, 7, -4], [18, -10, 3, 9], [-1, -6, 2, -15], [0, -14, -4, 4], [2, -25, -7, -3], [10, 24, 4, 0], [0, -10, -9, 3], [-8, 8, 4, -8], [28, -6, 2, -10], [22, -19, 3, 9], [-10, 11, -3, 13], [16, -10, -2, 6], [-2, 4, 3, -19], [-11, 22, 2, -1], [-4, 17, -2, -2], [-28, 4, 0, 8], [-10, 5, 1, 17], [4, -14, 6, 10], [2, -14, -6, -12], [-10, -14, -2, 0], [-27, 2, -1, 4], [-4, 2, -6, 14], [-16, 4, 5, -3], [-10, -12, -12, 0], [5, 2, 2, 15], [-2, 1, -11, -15], [-19, -20, 8, -3], [-8, 4, 0, 4], [2, 5, 9, 9], [11, 7, -6, 9], [-12, 2, 9, -7], [-2, 2, -12, -10], [-22, -8, -4, 0], [18, -6, 4, 6], [-4, 2, 1, -19], [8, 10, -10, 6], [-17, 13, 10, 5], [38, 13, 1, -3], [-19, 12, -8, -11], [-22, 4, 10, 4], [-3, -16, -4, 9], [28, 20, 14, -4], [-28, -8, 4, -4], [2, -3, -3, 1], [10, 4, -6, 4], [2, 5, 6, -4], [-14, 3, -1, -5], [22, -11, 3, 1], [-35, -10, 4, 1], [4, -18, 6, 6], [-22, 8, -3, -5], [-6, 7, 7, -9], [-2, -31, 1, 5], [-29, 2, -4, 7], [18, -14, 8, 10], [0, -2, -14, -2], [-44, -8, -2, -2], [23, 0, 4, 3], [15, 2, -6, -7], [28, 2, -6, -6], [5, 22, -2, 3], [9, 23, -8, 9], [12, -18, 10, 10], [26, -1, 3, -13], [-16, 14, 2, -10], [34, 13, 1, 5], [7, 22, -8, 7], [4, 36, -4, 0], [-4, 2, 0, 4], [-44, -10, 0, 0], [-10, -9, 16, -2], [-2, -1, -11, 3], [-25, -8, 8, 7], [6, -4, -10, 12], [2, 20, 0, -12], [-12, -24, 2, -2], [12, -6, -4, 12], [12, -18, 0, 16], [20, -22, -6, -2], [-21, -9, 4, 3], [6, -12, -16, -12], [20, 14, 1, -3], [22, 2, -4, 2], [-10, 18, -12, -14], [2, 17, 5, 1], [2, 27, -5, 7], [-7, -2, 1, -10], [-36, 16, 0, 12], [26, 9, 1, 1], [6, -14, -10, -12], [-12, -12, 2, 2], [-44, 3, 2, 6], [-4, 10, -10, 14], [-8, -18, -6, -6], [-29, 8, -9, -2], [-2, 13, 3, 17], [-25, 12, -3, 4], [1, 11, -8, -7], [14, -6, -2, -10], [5, 24, 8, 5], [30, 16, -14, 0], [33, -11, -6, -9], [12, 12, -2, 2], [-18, 6, 12, -2], [26, 9, -9, 5], [-20, -8, 10, -8], [-11, -14, 4, -7], [6, 2, -12, 10], [24, 3, 10, -14], [-12, -1, 0, 12], [46, 9, 1, 5], [30, 5, -7, 13], [5, 4, 3, 8], [-15, -4, -4, 5], [-31, -2, -2, -1], [-20, 16, -8, -4], [0, 0, 16, -12], [14, 14, 3, -11], [-8, 0, -3, -19], [4, -2, -9, -9], [6, 0, 2, -12], [26, -27, 9, 13], [-24, -6, 7, 7], [-2, 14, 20, 14], [-22, 18, -4, -10], [-15, -10, 2, -13], [32, -16, 0, 0], [-46, -7, -8, 2], [-46, 3, 1, -1], [19, 14, -6, -3], [7, 19, -14, -7], [28, 2, 2, -6], [28, 20, 14, -4], [-10, 10, 10, 6], [-19, -11, -4, 13], [26, 12, 0, -2], [0, 18, 2, 10], [-10, 24, -4, -10], [-12, -10, 19, 11], [33, -6, -3, 6], [14, -17, 5, 5], [30, 16, -10, 4], [12, -8, -16, 4], [15, 3, 14, -7], [-1, -28, 4, -5], [-2, -21, 3, -13], [-30, 22, 6, 8], [26, -2, -14, -2], [-9, 31, -8, -5], [-14, -8, 4, 14], [10, 5, 0, 18], [-3, 37, -2, -9], [-31, 28, 0, 9], [-51, 1, 0, -7], [-19, -20, 3, 0], [-5, -6, -8, 11], [-38, -16, 4, 0], [-42, -7, 7, 1], [38, 5, 7, 1], [-16, 30, 2, 14], [20, -34, 0, 14], [-28, 0, -8, -4], [41, -12, 6, 15], [10, 8, 18, 4], [18, -8, -8, -6], [6, -30, -2, 0], [54, 10, -6, 6], [38, 6, -14, 2], [30, -3, -5, -3], [20, 12, 0, -8], [-5, 4, -8, 11], [-30, 16, -6, 0], [14, -7, 19, 5], [2, 35, 8, 6], [10, -8, -16, -4], [-11, -24, -9, -4], [-12, -16, 8, 12], [-16, 1, 4, -4], [-10, 21, -3, -7], [22, -15, -3, -15], [2, -5, -8, 2], [-8, -2, -2, 6], [-28, -4, 16, 0], [-38, -7, -1, 13], [-2, 0, -4, -8], [-44, 12, -2, -12], [-18, 6, -2, -2], [-8, -16, 6, 2], [-3, -9, -10, 7], [-14, -8, 6, 12], [-24, -18, 10, -10], [34, 5, 1, -3], [-15, -6, 14, -17], [-40, -13, -4, 0], [-34, 0, 16, 10], [-34, -6, 6, -2], [2, 14, -6, 12], [-16, 7, -4, 0], [-42, 9, 5, 13], [22, -14, -2, -18], [12, -12, 4, -28], [13, 6, -8, -19], [8, 20, 0, 8], [-8, -2, 2, 10], [27, -4, 18, 13], [-61, -16, -3, 0], [-6, -28, 2, 8], [34, 5, -11, -7], [4, 34, -2, 10], [-30, 16, 0, -10], [-19, -20, -4, -7], [33, 8, -3, 6], [-5, -19, -2, -11], [14, 21, 17, -11], [-14, 12, 22, 8], [38, 12, 12, -2], [1, 16, 20, 5], [14, -16, -14, -16], [16, 26, -6, -2], [-18, -16, -8, 6], [-8, -6, -2, -14], [28, 19, -12, 0], [14, 13, 27, 5], [-14, -12, -16, -8], [4, 8, -16, -16], [42, 28, 8, -6], [34, -29, 1, -9], [-16, 12, -2, 20], [-8, 4, -12, -28], [6, 14, -10, 18], [40, 4, 8, 12], [-30, 1, -5, 13], [9, -14, 0, 17], [-9, 28, -2, 13], [-13, 14, -10, 17], [29, -24, 14, 19], [-2, -6, -6, -6], [-22, 23, 5, -5], [-14, -1, 11, 23], [-11, 30, 1, 6], [17, 8, 9, -6], [22, -11, -11, -7], [19, 28, -10, -7], [-10, 31, 1, -11], [-23, -2, -13, 8], [-16, -10, 22, 2], [59, -6, -4, 7], [32, -18, -2, -6], [-30, -14, 16, -2], [2, -2, -16, 18], [10, -15, -3, -7], [20, -36, 8, 12], [12, -32, -3, 13], [-22, -25, -15, -9], [30, -8, -4, -16], [2, -34, -1, -3], [22, -20, 4, -4], [0, -6, 18, 6], [24, -8, -17, -5], [-24, 27, -10, -2], [-4, -16, 2, -6], [26, -2, 6, 6], [-26, -23, 17, 5], [24, 14, -14, 14], [-12, 16, 16, 24], [11, -18, -6, -3], [-26, 10, 8, 2], [36, -2, 14, 14], [-42, 10, 12, 14], [34, 21, 17, -11], [-31, 6, -12, -11], [18, -5, -3, -1], [29, -1, -12, -3], [-12, -8, 19, 7], [-12, -16, -4, 16], [-38, -6, 6, -18], [-6, -10, 11, -23], [10, 14, 8, 2], [25, -17, -14, -13], [34, 3, 22, 16], [11, -30, -8, 3], [-18, -15, -21, 9], [14, -8, 8, -8], [12, -14, 10, 2], [22, 24, 4, 6]];
    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_bl();".
// 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_bl(: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_bl();".
// 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_bl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3234_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![8, -16, -4, 4, 1]>,<13,R![-28, -80, -4, 8, 1]>,<17,R![776, -112, -52, 4, 1]>],Snew);
    return Vf;
end function;