
// Make newform 2640.2.a.f in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2640_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_2640_2_a_f();".
// 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_2640_2_a_f();".
// 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_2640_a();"
function MakeCharacter_2640_a()
    N := 2640;
    order := 1;
    char_gens := [991, 661, 881, 1057, 1201];
    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_2640_a_Hecke(Kf)
    return MakeCharacter_2640_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], [-1], [-1], [4], [1], [-4], [-6], [-2], [0], [0], [4], [-10], [0], [4], [-12], [6], [-12], [-10], [4], [0], [8], [10], [6], [-6], [-10], [-12], [16], [-18], [14], [-6], [-8], [0], [6], [-14], [0], [-14], [-22], [-20], [-18], [-18], [-12], [2], [0], [-4], [-18], [16], [22], [-8], [18], [26], [-6], [0], [-22], [-12], [-18], [18], [6], [-2], [-28], [0], [4], [30], [16], [24], [-22], [6], [28], [8], [6], [14], [-30], [36], [16], [-4], [-8], [12], [18], [26], [30], [-10], [36], [26], [36], [38], [10], [-24], [-30], [-28], [-24], [-32], [-12], [0], [16], [-24], [4], [-42], [-42], [6], [16], [14], [-8], [-6], [-18], [24], [-2], [-34], [-36], [6], [0], [-10], [40], [44], [-6], [4], [-8], [-30], [4], [36], [18], [-36], [38], [44], [42], [-24], [-32], [12], [-10], [24], [-8], [32], [22], [6], [-8], [-22], [-24], [14], [-6], [-20], [-18], [36], [10], [24], [-32], [-18], [-34], [24], [-28], [-6], [-20], [-36], [-4], [30], [-20], [-6], [28], [-24], [-38], [6], [8], [12], [48], [30], [-8], [36], [30], [36], [-44], [44], [26], [54], [-12], [-58], [12], [20], [40], [-42], [22], [30], [16], [-46], [16], [-12], [-22], [-18], [12], [6], [8], [-20], [-58], [60], [-34], [54], [52], [66], [-18], [-42], [14], [-10], [-18], [-66], [60], [-26], [14], [50], [-60], [-18], [-20], [30], [36], [40], [32], [54], [40], [-12], [-12], [-10], [-32], [-12], [-48], [-30], [-22], [-14], [30], [-32], [66], [26], [-42], [72], [4], [-36], [26], [-14], [-50], [-12], [52], [54], [14], [54], [-12], [48], [0], [58], [-32], [-46], [-6], [12], [-56], [60], [-26], [6], [-4], [-60], [-48], [-10], [-6], [-24], [-10], [-44], [30], [8], [64], [30], [-70], [56], [-30], [-20], [-18], [30], [-56], [-78], [-46], [28], [-34], [10], [20], [-32], [24], [-34], [-10], [-24], [18], [-44], [-42], [-70], [-32], [24], [-34], [42], [4], [60], [-30], [72], [-78], [72], [8], [48], [-8], [66], [48], [-8], [32], [-54], [-2], [-36], [-32], [-22], [48], [-58], [-72], [-52], [-42], [-6], [-12], [4], [30], [50], [36], [-12], [-10], [24], [22], [-58], [12], [-8], [-54], [-58], [88], [-44], [42], [6], [26], [-30], [10], [18], [34], [-72], [-94], [66], [-82], [52], [26], [42], [0], [-32], [18], [-84], [38], [52], [24], [-30], [10], [2], [30], [-8], [14], [18], [0], [-60], [30], [-48], [8], [-84], [-48], [-36], [52], [38], [18], [52], [38], [-12], [52], [6], [36], [-14], [-70], [12], [-72], [-16], [-48], [44], [90], [6], [-32], [66], [-14], [-48], [-8], [-22], [88], [-96], [-22], [6], [12], [-92], [-24], [-52], [94], [-78], [-8], [0], [-22], [-6], [-44], [54], [-48], [-38], [-46], [-36], [28], [84], [-16], [102], [-36], [-26], [8], [-78], [36], [-80], [-78], [30], [6], [104], [-48], [-36], [50], [-54], [12], [-12], [16], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2640_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2640_2_a_f();".
// 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_2640_2_a_f(:prec:=1)
    chi := MakeCharacter_2640_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_2640_2_a_f();".
// 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_2640_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2640_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, 1]>,<13,R![4, 1]>,<17,R![6, 1]>,<19,R![2, 1]>],Snew);
    return Vf;
end function;