
// Make newform 3234.2.a.ba 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_ba();".
// 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_ba();".
// 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 := [-7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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], [1, 0], [0, 1], [0, 0], [1, 0], [4, 0], [3, 0], [0, 2], [0, 1], [2, 0], [4, 0], [-4, -2], [-9, 0], [4, -2], [-4, -3], [4, 0], [4, 4], [-4, 3], [-3, -4], [-8, -2], [10, -2], [8, -1], [-5, 4], [-8, -2], [1, -4], [-2, 2], [10, 0], [9, 0], [4, 3], [2, -4], [0, -1], [8, -4], [-8, 2], [2, -4], [4, -6], [20, -1], [6, 2], [5, 4], [-14, -2], [-4, 0], [10, 2], [0, -2], [12, 2], [0, 6], [6, -6], [-2, -8], [16, -2], [-2, 10], [-13, -4], [-12, -2], [-1, -8], [8, 2], [6, -8], [-18, 2], [-4, 2], [6, -6], [8, -1], [-4, 2], [4, 8], [-3, 0], [-4, 10], [4, -4], [22, 4], [-4, 7], [10, -4], [-4, -5], [13, 0], [12, 2], [19, -4], [0, -5], [18, -2], [10, -4], [-16, -2], [8, 9], [-17, 4], [-24, -2], [24, -3], [-6, 0], [-20, 6], [-20, 6], [-6, -6], [22, 6], [-12, 2], [9, 4], [12, 1], [-10, 8], [4, -10], [-30, 4], [14, -4], [2, -14], [-8, 0], [18, -6], [6, -6], [-1, 0], [-8, 4], [-4, -2], [-4, 8], [-42, 0], [-8, -2], [-20, -9], [20, 2], [-14, 2], [0, 0], [6, 8], [2, 12], [9, 12], [6, 6], [-6, -12], [32, 1], [28, -4], [-16, -11], [-16, -9], [-20, -2], [41, 0], [26, -6], [-16, 4], [-20, -8], [-8, -13], [-16, 11], [-3, 8], [-26, -2], [-30, -4], [24, 6], [4, 8], [7, 8], [16, -10], [-16, -12], [36, 3], [2, 0], [-4, 3], [-14, 4], [-24, -2], [-24, 2], [-22, 4], [-37, 4], [36, 2], [16, 3], [-4, 2], [-16, 11], [3, -16], [-6, 4], [12, 16], [-28, -6], [-7, -16], [-4, 6], [8, -11], [4, 7], [-31, -8], [-17, 12], [-4, 11], [24, 5], [-2, 8], [-45, -4], [-6, -14], [-37, -8], [16, 13], [12, 1], [8, -10], [-50, 0], [-14, -4], [-26, -4], [-5, -12], [48, 3], [-6, 0], [-8, -8], [-24, 7], [-28, 6], [-28, -8], [12, 2], [-12, 0], [9, 4], [-12, -2], [10, 8], [-52, -2], [14, -18], [-16, 2], [-6, 0], [-40, 8], [-16, -9], [-8, 0], [32, 2], [35, -4], [36, -2], [7, 8], [24, -6], [24, 4], [40, 9], [11, 8], [4, 12], [-20, -2], [30, -8], [-3, 12], [27, -8], [32, 4], [-4, 16], [18, 12], [-34, -4], [-28, -8], [2, 8], [20, -16], [10, 4], [8, -2], [8, 2], [24, -8], [-10, -4], [24, 4], [20, 8], [5, -12], [27, 12], [-48, 0], [22, -16], [16, -19], [-22, 6], [42, 6], [-28, 10], [-22, 0], [-8, 6], [13, -4], [24, 5], [24, 12], [-24, 4], [-12, -7], [-24, -6], [8, 0], [-25, -8], [24, 1], [-6, 6], [24, -9], [-36, -11], [-24, -12], [-12, 4], [42, 2], [56, -6], [-13, 8], [11, -12], [-22, -14], [-30, -4], [-2, -2], [20, 2], [-4, -27], [-4, -4], [40, -4], [48, -4], [50, 0], [-2, -8], [36, 6], [6, 4], [-18, 4], [22, -10], [40, -8], [-48, 4], [34, -8], [20, -1], [-51, -4], [-34, 0], [4, 24], [18, -2], [-6, 16], [-40, -4], [10, -2], [-12, 9], [8, -20], [-28, -9], [-20, 17], [-26, -14], [-12, 0], [4, 1], [-4, 16], [-4, 26], [-26, -14], [-22, 10], [27, 8], [22, -8], [20, 3], [2, -18], [16, 2], [-8, 14], [8, -9], [42, 2], [13, -12], [-6, 12], [-46, 8], [32, 4], [0, 11], [-26, 14], [8, -11], [30, -8], [-46, 6], [14, 0], [50, 4], [4, 1], [4, -18], [37, -4], [7, 20], [-60, 7], [-12, 10], [26, 0], [28, -1], [-23, -12], [-22, -16], [6, 14], [14, 4], [0, 26], [-66, 4], [-25, 8], [15, 4], [4, 8], [-42, 10], [8, 2], [-36, -7], [22, -4], [-40, 4], [-3, -24], [25, 24], [34, 2], [-20, 26], [-10, 24], [-20, -4], [-31, 4], [-26, -4], [-58, 8], [-39, -4], [22, -2], [-24, 4], [-10, 8], [-26, 0], [-1, -12], [-5, -4], [80, 4], [-20, -4], [0, 19], [24, 24], [48, -2], [13, -24], [40, -12], [12, -18], [-16, -20], [-38, 12], [-46, -12], [32, 6], [24, 16], [20, 10], [60, 12], [-6, 8], [-48, 8], [24, -12], [38, -16], [0, -20], [64, -4], [60, -9], [-60, -2], [-17, -8], [-16, -1], [24, 9], [-4, -24], [-10, -8], [32, -1], [29, 0], [10, -28], [-92, -1], [-8, 0], [-17, 0], [48, -6], [-63, -12], [32, 4], [-31, 12], [-2, -4], [-48, -2], [-10, 20], [8, 6], [49, -4], [-34, 0], [68, -2], [-4, -2], [-8, 16], [46, -10], [12, 6], [-2, -10], [-59, -8], [54, -10], [8, 5], [-68, 0], [48, 15], [-30, -12], [-8, 4], [-28, -15], [54, -4], [36, 10], [-24, 24], [-20, 17], [-23, 0], [20, -25], [38, 12], [-7, -12], [-28, -13], [-6, -14], [-28, -6], [88, 2], [32, -12], [-12, 14], [8, -20], [62, -2], [-36, -10], [-42, -6], [-8, 10], [64, 7], [2, -4], [69, 12], [73, 0], [-60, 6], [-46, -20], [14, 6], [-4, 24], [-8, 8], [-32, -8], [-8, 15], [38, -16], [-42, 20], [34, -10], [-12, 10], [76, 7], [-16, 13], [40, -14], [16, -12], [78, -4], [-72, 13], [-38, -6], [46, 8], [1, 0], [-46, 10]];
    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_ba();".
// 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_ba(:prec:=2)
    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_ba();".
// 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_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3234_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-7, 0, 1]>,<13,R![-4, 1]>,<17,R![-3, 1]>],Snew);
    return Vf;
end function;