
// Make newform 5586.2.a.z in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5586_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_5586_2_a_z();".
// 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_5586_2_a_z();".
// 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_5586_a();"
function MakeCharacter_5586_a()
    N := 5586;
    order := 1;
    char_gens := [3725, 4903, 4999];
    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_5586_a_Hecke(Kf)
    return MakeCharacter_5586_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], [-2], [2], [-2], [1], [-6], [-6], [0], [-10], [8], [0], [0], [-2], [-12], [8], [-2], [-4], [-8], [2], [-4], [0], [6], [-2], [8], [-4], [-2], [-10], [22], [-4], [-10], [-12], [-6], [2], [-12], [-8], [4], [-24], [-12], [14], [-6], [10], [18], [-20], [-10], [-8], [8], [8], [-18], [18], [-6], [-20], [-4], [30], [24], [-8], [-10], [-6], [20], [0], [12], [0], [8], [-30], [-14], [22], [18], [-4], [6], [6], [24], [-14], [-6], [-24], [-10], [20], [-18], [-2], [20], [34], [-12], [30], [8], [6], [-2], [38], [22], [-8], [20], [24], [-42], [-14], [-16], [-8], [0], [0], [36], [34], [-26], [42], [-28], [6], [20], [-32], [12], [-14], [36], [30], [0], [-14], [14], [-24], [4], [10], [-8], [24], [46], [20], [-34], [14], [-20], [16], [28], [6], [-10], [-40], [-8], [44], [-4], [24], [-10], [18], [-22], [16], [-24], [-36], [-48], [2], [-20], [-38], [0], [-28], [34], [12], [-8], [32], [56], [0], [-2], [-34], [8], [-8], [-6], [-20], [40], [-42], [28], [24], [18], [-42], [-44], [-16], [-18], [-60], [18], [-4], [10], [-50], [16], [-10], [30], [38], [-56], [-54], [-60], [46], [32], [-8], [20], [0], [-26], [44], [22], [-50], [-38], [-4], [-50], [-32], [-46], [-6], [22], [36], [26], [-56], [14], [-22], [30], [-40], [-26], [8], [-26], [18], [20], [-34], [48], [24], [-6], [-28], [-38], [22], [-56], [32], [60], [10], [40], [0], [28], [-18], [-50], [32], [-46], [56], [48], [10], [-20], [24], [-8], [-54], [-18], [20], [-10], [-38], [-44], [0], [40], [-6], [-74], [0], [24], [20], [4], [-46], [44], [64], [72], [-64], [66], [14], [2], [18], [2], [-14], [18], [14], [-10], [44], [-12], [28], [46], [-72], [20], [66], [-22], [-4], [-42], [-30], [-66], [-34], [22], [-38], [-64], [-4], [70], [20], [54], [-22], [34], [52], [-36], [-32], [-48], [2], [52], [46], [-34], [38], [60], [-36], [-30], [-76], [74], [12], [-46], [-6], [-8], [-50], [-36], [-40], [-4], [26], [-60], [-64], [40], [14], [-72], [-54], [34], [26], [56], [-10], [-22], [-2], [18], [62], [-12], [-12], [-84], [30], [60], [10], [-4], [-6], [-46], [66], [-46], [44], [84], [-66], [-78], [-66], [-88], [-4], [-48], [-36], [-2], [-40], [-84], [8], [-54], [54], [-84], [46], [-66], [20], [-60], [22], [92], [78], [-92], [-86], [-38], [72], [-86], [-48], [8], [12], [42], [44], [90], [-78], [-4], [-24], [60], [-34], [-78], [70], [26], [22], [-92], [-74], [-54], [-8], [82], [-72], [42], [-16], [-38], [-62], [60], [-30], [76], [2], [-68], [56], [4], [64], [-72], [60], [-90], [60], [-70], [-56], [16], [22], [-8], [84], [10], [30], [50], [10], [74], [-84], [0], [0], [46], [50], [-44], [92], [-82], [-34], [-12], [-48], [-14], [66], [0], [0], [22], [-24], [-66], [-2], [46], [12], [86], [0], [-28], [-10], [-80], [56]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5586_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5586_2_a_z();".
// 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_5586_2_a_z(:prec:=1)
    chi := MakeCharacter_5586_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_5586_2_a_z();".
// 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_5586_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5586_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<11,R![2, 1]>,<13,R![-2, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;