
// Make newform 3234.2.a.q in Magma, downloaded from the LMFDB on 24 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_q();".
// 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_q();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
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], [-1], [0], [0], [-1], [-2], [0], [-2], [0], [-6], [-2], [2], [0], [-4], [6], [-6], [0], [-2], [-4], [-12], [4], [8], [-6], [6], [-2], [-6], [-14], [0], [-10], [-18], [8], [6], [-18], [-14], [6], [8], [16], [20], [-12], [-18], [-12], [-20], [24], [2], [-6], [10], [20], [-2], [-18], [-20], [18], [0], [28], [0], [-18], [24], [0], [-20], [-10], [-6], [-14], [-18], [34], [-6], [10], [18], [-4], [14], [0], [10], [-18], [0], [22], [2], [20], [-6], [6], [16], [6], [40], [12], [-10], [0], [22], [28], [-12], [-18], [-22], [-6], [8], [36], [-24], [-4], [-12], [-4], [36], [24], [6], [10], [2], [8], [18], [-18], [-6], [32], [10], [24], [-24], [-36], [-8], [40], [26], [42], [-20], [20], [-30], [-32], [-6], [30], [0], [-44], [14], [-18], [-12], [-8], [-18], [26], [18], [46], [-26], [-16], [48], [-4], [26], [12], [16], [-36], [-50], [-36], [-42], [10], [-42], [20], [36], [52], [-6], [-2], [-36], [40], [-24], [14], [-30], [44], [24], [-52], [0], [-16], [6], [52], [-18], [-12], [-42], [-16], [12], [18], [6], [-40], [10], [-22], [-48], [-24], [-32], [-48], [62], [-38], [18], [8], [-42], [52], [34], [8], [-42], [-22], [-36], [60], [-24], [-22], [-44], [-34], [-12], [-62], [-24], [20], [-36], [60], [6], [-10], [2], [24], [-36], [30], [52], [16], [4], [-12], [-24], [-62], [-60], [-42], [4], [38], [-60], [-16], [0], [60], [34], [-16], [60], [12], [-42], [-22], [4], [30], [8], [-42], [-10], [-6], [60], [28], [-24], [50], [-26], [-64], [30], [28], [-60], [-50], [-42], [12], [54], [-36], [-26], [-14], [14], [-12], [-60], [10], [48], [-40], [-24], [-22], [72], [-12], [-26], [78], [24], [50], [-38], [-72], [-68], [56], [36], [22], [34], [-6], [-8], [-54], [-6], [8], [-6], [40], [-52], [10], [8], [-20], [46], [-36], [2], [50], [-6], [72], [-40], [24], [-38], [-2], [-24], [-58], [-18], [10], [-12], [-54], [12], [-6], [-30], [-82], [18], [-62], [72], [78], [10], [16], [-18], [8], [84], [-76], [62], [-12], [-68], [36], [38], [-48], [-18], [-54], [-4], [0], [-20], [-48], [48], [-62], [18], [-74], [-70], [-42], [-76], [30], [-38], [20], [-44], [48], [6], [26], [78], [-44], [54], [-88], [12], [26], [36], [-50], [-56], [14], [-54], [78], [-28], [-66], [84], [52], [-52], [-96], [36], [58], [-10], [42], [-8], [-58], [84], [30], [-42], [54], [36], [86], [-12], [-12], [36], [4], [-46], [-42], [8], [26], [-60], [-20], [-48], [42], [-8], [-34], [-60], [24], [40], [24], [28], [-36], [66], [56], [-66], [-38], [42], [-100], [4], [-4], [54], [74], [36], [-60], [64], [12], [14], [-56], [-54], [-52], [-42], [94], [18], [-40], [-30], [102], [100], [-58], [6], [28], [60], [-44], [-30], [-36], [32], [-58], [36], [72], [46], [42], [24], [-18], [70], [36], [-30], [82], [12], [36], [66], [-92], [-24]];
    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_q();".
// 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_q(:prec:=1)
    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_q();".
// 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_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3234_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<13,R![2, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;