
// Make newform 5586.2.a.n in Magma, downloaded from the LMFDB on 23 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_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_5586_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_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], [0], [0], [0], [0], [-4], [-1], [-4], [2], [8], [10], [-8], [-4], [8], [-10], [-4], [4], [-8], [0], [-16], [-12], [-4], [0], [-4], [0], [8], [4], [-14], [18], [20], [-20], [-10], [4], [-10], [-12], [20], [4], [-24], [-4], [-20], [8], [-4], [2], [-10], [8], [8], [16], [12], [12], [6], [-12], [-12], [-12], [8], [-4], [-20], [0], [-10], [6], [-20], [-12], [12], [0], [-8], [-2], [0], [30], [-24], [20], [20], [4], [-16], [-26], [-16], [8], [10], [20], [-2], [28], [20], [-26], [-16], [28], [16], [-40], [-30], [-6], [-8], [-24], [28], [24], [-20], [16], [-20], [-16], [-36], [-40], [-20], [-2], [16], [-2], [4], [22], [-28], [16], [-12], [12], [0], [4], [32], [-6], [-42], [28], [-40], [-2], [20], [8], [34], [-12], [-32], [14], [-12], [-4], [-4], [-14], [26], [24], [40], [-4], [12], [-48], [20], [10], [52], [24], [20], [-52], [28], [26], [28], [6], [16], [52], [-56], [-40], [44], [-24], [20], [-16], [34], [-28], [-20], [-56], [8], [-48], [-32], [44], [-56], [-12], [48], [-6], [32], [-36], [30], [24], [-44], [-52], [18], [24], [44], [8], [12], [22], [8], [60], [28], [42], [0], [-12], [-8], [-12], [-38], [24], [44], [-8], [30], [-52], [22], [8], [-36], [48], [-8], [20], [-8], [32], [-50], [2], [-60], [-64], [46], [-8], [-48], [28], [-36], [48], [40], [60], [-54], [4], [62], [0], [56], [-4], [-16], [20], [8], [56], [-64], [2], [26], [16], [-46], [48], [-36], [10], [72], [-8], [32], [-56], [50], [36], [-20], [-6], [-4], [48], [8], [54], [40], [16], [44], [12], [64], [-2], [72], [40], [-32], [28], [-16], [36], [-34], [-20], [-28], [60], [-8], [0], [74], [-12], [12], [-16], [4], [12], [-4], [-72], [-12], [-28], [46], [28], [-32], [26], [-40], [-56], [-16], [-24], [-60], [-64], [32], [46], [42], [12], [0], [-16], [-72], [40], [-4], [-36], [34], [54], [-40], [-40], [-18], [76], [6], [-68], [14], [48], [-32], [72], [68], [4], [-16], [18], [16], [-36], [-36], [-18], [-12], [-56], [12], [-58], [0], [42], [-34], [-64], [-12], [68], [-28], [24], [8], [-50], [-44], [22], [76], [20], [-70], [-12], [-8], [44], [-48], [-54], [14], [-30], [40], [-12], [-12], [-68], [50], [-64], [-40], [-56], [22], [54], [52], [20], [66], [4], [36], [8], [32], [48], [4], [-22], [46], [-72], [6], [-80], [-40], [-36], [2], [72], [70], [28], [48], [44], [76], [-10], [80], [-4], [-54], [-48], [-60], [-76], [-74], [0], [-66], [-36], [4], [32], [-12], [52], [68], [26], [-88], [66], [-76], [-32], [16], [-28], [-44], [24], [-50], [92], [-48], [4], [-96], [-90], [-88], [48], [40], [10], [-104], [-78], [-28], [16], [44], [104], [78], [66], [-12], [36], [4], [-38], [-84], [-20], [38], [48], [-32], [8], [76], [-48], [-30], [32], [100], [20], [12], [-76], [-76], [22], [-92], [40]];
    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_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_5586_2_a_n(: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_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_5586_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5586_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![0, 1]>,<13,R![0, 1]>,<17,R![4, 1]>],Snew);
    return Vf;
end function;