
// Make newform 3640.2.a.g in Magma, downloaded from the LMFDB on 19 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_g();".
// 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_g();".
// 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], [-2], [-1], [6], [-6], [6], [-6], [-6], [6], [-10], [-6], [0], [6], [-10], [2], [8], [-10], [-14], [-12], [12], [-10], [-2], [-2], [-6], [2], [-2], [-6], [2], [12], [-10], [0], [22], [2], [10], [-24], [0], [-14], [-20], [10], [8], [-10], [18], [0], [4], [8], [8], [10], [2], [-22], [30], [-16], [-18], [18], [-26], [-2], [14], [26], [14], [14], [12], [-20], [14], [30], [-14], [10], [26], [2], [-30], [-18], [10], [30], [22], [0], [-26], [-2], [-6], [-34], [-24], [14], [-10], [6], [32], [6], [-6], [14], [42], [-4], [-30], [-30], [-28], [24], [-34], [-34], [42], [6], [10], [6], [-22], [-18], [-26], [22], [-24], [26], [8], [-34], [-36], [-22], [2], [26], [-2], [-10], [-6], [34], [44], [6], [30], [-40], [-46], [26], [18], [16], [14], [-18], [22], [-16], [-34], [2], [2], [-24], [-20], [-34], [-34], [14], [-26], [-4], [-22], [-2], [-10], [38], [14], [20], [34], [34], [46], [-50], [8], [-16], [-22], [14], [2], [34], [54], [56], [-20], [-18], [22], [10], [-4], [-22], [4], [-20], [18], [36], [56], [-22], [34], [-6], [-42], [58], [60], [-6], [40], [-30], [-18], [-42], [-54], [-10], [16], [0], [-42], [6], [44], [-22], [-34], [-28], [10], [-18], [6], [36], [-52], [-62], [62], [30], [-22], [-2], [-46], [-58], [-66], [8], [10], [46], [-42], [18], [14], [46], [58], [0], [58], [-22], [-6], [-60], [-18], [-18], [22], [34], [-52], [-6], [58], [34], [-54], [72], [58], [-6], [-50], [-12], [-2], [-10], [54], [-72], [-18], [-6], [26], [-28], [14], [-10], [-44], [48], [-60], [-32], [14], [-42], [-18], [-56], [8], [-78], [62], [6], [-2], [46], [-8], [26], [18], [-6], [10], [-8], [-54], [-30], [-18], [-30], [-78], [-14], [-78], [32], [46], [-50], [-48], [-34], [-42], [-16], [-26], [-32], [-2], [-8], [72], [-18], [74], [-80], [-34], [58], [-22], [-70], [-60], [-60], [-22], [-78], [-78], [30], [18], [-2], [54], [2], [62], [-18], [-80], [-14], [72], [-60], [46], [2], [-88], [-50], [12], [82], [-34], [54], [-42], [46], [6], [-66], [14], [-58], [24], [-74], [82], [38], [-70], [18], [-16], [26], [2], [-76], [26], [-34], [-30], [-44], [2], [22], [70], [38], [-28], [8], [-78], [48], [-50], [-50], [30], [-74], [58], [90], [-62], [12], [-30], [-36], [38], [68], [22], [90], [70], [22], [-74], [74], [22], [-10], [18], [6], [-70], [-16], [22], [-42], [-42], [70], [66], [26], [50], [-52], [-54], [-68], [-28], [36], [-78], [16], [-10], [18], [100], [-18], [78], [-90], [-30], [-6], [64], [2], [-74], [-44], [-10], [50], [-20], [26], [-22], [-10], [66], [-2], [-46], [-22], [-34], [-26], [-100], [-10], [66], [-14], [80], [10], [90], [-32], [-2], [66], [-64], [46], [-74], [-74], [6], [72], [58], [30], [-6], [-54], [98], [-30], [2], [42], [4], [0], [-90], [70], [50], [-78], [10], [-84]];
    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_g();".
// 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_g(: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_g();".
// 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_g( : 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![2, 1]>],Snew);
    return Vf;
end function;