
// Make newform 3640.2.a.j in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3640_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_3640_2_a_j();".
// 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_3640_2_a_j();".
// 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_3640_a();"
function MakeCharacter_3640_a()
    N := 3640;
    order := 1;
    char_gens := [911, 1821, 1457, 521, 561];
    v := [1, 1, 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_3640_a_Hecke(Kf)
    return MakeCharacter_3640_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 := [[0], [2], [1], [-1], [0], [-1], [-8], [-2], [-6], [-2], [-4], [2], [-8], [8], [-8], [0], [-14], [2], [12], [2], [2], [8], [16], [0], [-2], [10], [-4], [16], [4], [2], [-18], [-16], [10], [8], [-4], [-6], [-22], [12], [-8], [2], [12], [6], [12], [-6], [2], [-8], [12], [8], [0], [-6], [-10], [-22], [-20], [-12], [8], [-6], [6], [20], [-12], [-18], [10], [6], [20], [-32], [16], [-18], [8], [-26], [12], [10], [-34], [-22], [16], [-16], [12], [24], [14], [-26], [-30], [8], [24], [-28], [-18], [-24], [-24], [32], [-18], [22], [-38], [-16], [18], [-24], [-24], [36], [-24], [4], [18], [-10], [10], [-36], [28], [-46], [-22], [26], [4], [-14], [-16], [6], [16], [18], [-20], [6], [-26], [-10], [10], [-18], [40], [16], [-36], [20], [-42], [34], [18], [-4], [18], [50], [-20], [16], [16], [-2], [-16], [12], [32], [36], [-28], [0], [-22], [28], [18], [26], [-18], [-12], [-22], [12], [46], [-28], [26], [-32], [-44], [-56], [42], [42], [-36], [24], [12], [-48], [28], [-52], [44], [54], [28], [14], [48], [-36], [6], [-24], [32], [-42], [-42], [2], [24], [-46], [44], [18], [-56], [46], [28], [56], [48], [2], [0], [-52], [12], [-26], [-48], [10], [-12], [24], [10], [-12], [16], [44], [-28], [-62], [12], [-52], [30], [-8], [22], [56], [-12], [-16], [26], [42], [-22], [70], [-64], [-44], [10], [-8], [2], [-58], [34], [20], [52], [48], [14], [42], [-4], [42], [-54], [28], [-18], [20], [14], [10], [36], [-28], [44], [-8], [-24], [-44], [26], [-42], [70], [-72], [0], [-18], [-12], [-24], [-36], [-52], [4], [-24], [74], [-8], [48], [-42], [12], [42], [-54], [28], [44], [-6], [-66], [40], [10], [-40], [-18], [66], [26], [-36], [-50], [-42], [-66], [-48], [28], [12], [36], [60], [-26], [-12], [-2], [-52], [-16], [-72], [-36], [32], [2], [-40], [-72], [10], [-80], [-50], [-8], [-36], [-58], [-46], [-32], [-66], [-6], [-54], [6], [10], [4], [46], [-40], [-6], [24], [52], [76], [-18], [-32], [0], [20], [22], [-72], [34], [-66], [-28], [12], [-12], [-78], [-40], [16], [-54], [-66], [30], [-14], [46], [-24], [-38], [-30], [-4], [6], [-58], [52], [-64], [-50], [40], [20], [-36], [-56], [-16], [88], [-52], [92], [-82], [92], [-44], [62], [42], [-30], [48], [22], [12], [74], [-44], [60], [-74], [58], [-26], [76], [92], [-50], [72], [-64], [-90], [78], [16], [-18], [34], [-78], [28], [2], [14], [18], [-8], [38], [-52], [76], [20], [-42], [-16], [32], [-50], [52], [-46], [-38], [28], [18], [-54], [8], [-26], [10], [48], [30], [-58], [-84], [48], [-18], [70], [24], [98], [2], [74], [16], [-18], [4], [-40], [-50], [-2], [-36], [78], [10], [40], [-34], [30], [52], [70], [64], [84], [20], [-20], [-82], [46], [54], [16], [-74], [64], [66], [-106], [-84], [48], [34], [58], [12], [54], [-70], [88]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3640_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3640_2_a_j();".
// 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_3640_2_a_j(:prec:=1)
    chi := MakeCharacter_3640_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_3640_2_a_j();".
// 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_3640_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3640_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;