
// Make newform 3234.2.a.n in Magma, downloaded from the LMFDB on 25 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_n();".
// 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_n();".
// 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], [2], [0], [1], [-2], [6], [4], [-4], [2], [4], [-2], [6], [0], [8], [-14], [-12], [14], [4], [12], [-6], [0], [0], [6], [14], [-10], [4], [12], [10], [-6], [16], [0], [-22], [-12], [10], [0], [-2], [-20], [0], [-10], [-4], [14], [4], [-6], [-6], [12], [0], [12], [0], [14], [-2], [24], [26], [-20], [-18], [0], [-30], [0], [26], [-18], [4], [14], [12], [-8], [14], [10], [-20], [-14], [-28], [-34], [14], [32], [28], [26], [36], [-16], [18], [-18], [-14], [-38], [12], [-26], [-8], [-34], [-16], [12], [-22], [26], [30], [40], [-20], [24], [0], [-28], [12], [16], [18], [6], [36], [10], [-24], [-46], [-24], [22], [32], [6], [36], [-18], [-44], [26], [-8], [-46], [-30], [-4], [-40], [10], [-36], [-40], [-6], [-20], [-18], [-14], [-18], [-28], [44], [2], [-10], [-24], [-28], [-34], [-48], [-40], [16], [-2], [-26], [-38], [-14], [-12], [18], [-50], [44], [-22], [32], [-36], [14], [16], [-42], [22], [-20], [-36], [-14], [-34], [28], [-16], [-44], [12], [-16], [30], [-14], [38], [44], [-2], [0], [12], [18], [32], [8], [-26], [34], [18], [-4], [14], [8], [58], [-28], [54], [-48], [18], [32], [-50], [-24], [0], [-34], [-10], [-60], [2], [-14], [-12], [-14], [48], [38], [36], [-52], [-14], [52], [-42], [-54], [-2], [-42], [0], [18], [-8], [6], [2], [36], [2], [12], [-4], [-66], [-12], [50], [-70], [32], [12], [-8], [62], [40], [38], [-68], [-6], [-22], [40], [-30], [-16], [40], [-70], [6], [36], [-16], [12], [-58], [-20], [-16], [-18], [44], [-8], [46], [-54], [60], [56], [12], [-52], [-44], [38], [-42], [72], [-28], [-52], [-16], [-24], [2], [-66], [36], [22], [-10], [-20], [14], [44], [42], [-6], [24], [76], [38], [62], [-18], [36], [66], [-26], [24], [-54], [-2], [52], [-66], [56], [2], [-76], [-28], [74], [10], [16], [-24], [24], [64], [-26], [20], [52], [-46], [-22], [-52], [-26], [66], [4], [-34], [8], [2], [14], [-28], [26], [64], [-12], [-14], [82], [88], [44], [36], [2], [-28], [14], [-36], [58], [-16], [-86], [6], [76], [8], [74], [-76], [32], [-26], [-34], [-60], [-38], [-42], [56], [-42], [70], [44], [92], [-32], [74], [2], [-54], [-40], [16], [40], [-12], [-2], [-26], [-10], [8], [-82], [82], [30], [64], [34], [-12], [-18], [-28], [56], [-78], [68], [-54], [-6], [-72], [-46], [-74], [48], [56], [38], [-76], [-62], [-50], [-36], [-60], [-68], [26], [-58], [-40], [-30], [-52], [-28], [80], [10], [40], [78], [-76], [0], [-94], [54], [34], [66], [-14], [-72], [6], [-36], [-40], [64], [-10], [-48], [8], [-62], [90], [-4], [-52], [-36], [50], [80], [-10], [-100], [58], [46], [-14], [-80], [6], [30], [-48], [-18], [78], [28], [36], [10], [-22], [68], [8], [42], [-38], [48], [76], [-34], [-88], [-54], [22], [36], [-80], [94], [-70], [52], [-34], [100], [64]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3234_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<13,R![2, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;