
// Make newform 5586.2.a.u in Magma, downloaded from the LMFDB on 08 May 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_u();".
// 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_u();".
// 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], [4], [-6], [-1], [-6], [6], [-2], [-4], [-6], [-4], [-6], [6], [12], [-14], [8], [0], [-14], [-10], [12], [6], [10], [0], [10], [12], [-16], [-18], [2], [-12], [-18], [-20], [12], [-10], [-2], [-4], [12], [-18], [12], [16], [18], [-10], [12], [-20], [-4], [22], [-12], [10], [-6], [-18], [-14], [-12], [-18], [-18], [6], [-20], [-10], [-18], [4], [6], [16], [-30], [10], [18], [8], [2], [24], [-2], [6], [-6], [4], [-4], [8], [-24], [24], [-26], [30], [-14], [0], [-28], [-12], [10], [-26], [24], [-30], [2], [-12], [-4], [36], [6], [38], [36], [-4], [-6], [6], [30], [-8], [-22], [32], [12], [36], [42], [-28], [-2], [0], [6], [-12], [22], [34], [2], [-30], [4], [32], [-6], [4], [6], [36], [-12], [4], [-10], [18], [-12], [28], [-48], [-10], [18], [-8], [-50], [-4], [-12], [-10], [-46], [-6], [-14], [6], [-20], [-18], [-42], [16], [-24], [32], [-12], [16], [-36], [-38], [-18], [28], [12], [44], [30], [20], [-12], [8], [12], [-16], [-6], [58], [-18], [-12], [54], [-28], [12], [-18], [48], [26], [-26], [-34], [-12], [12], [40], [-18], [38], [-14], [30], [-4], [60], [-2], [34], [-28], [36], [32], [18], [6], [-36], [-16], [52], [-58], [6], [-2], [36], [32], [18], [0], [42], [14], [26], [-42], [54], [54], [-62], [4], [-50], [-24], [-24], [16], [-60], [66], [28], [50], [24], [20], [-36], [60], [-38], [-40], [18], [-60], [-24], [-40], [22], [-6], [32], [-60], [-10], [-42], [-12], [10], [0], [2], [4], [14], [42], [-44], [-42], [-14], [-48], [60], [0], [12], [52], [16], [-40], [-18], [42], [16], [60], [20], [18], [-22], [-42], [78], [46], [36], [48], [14], [16], [18], [58], [26], [12], [46], [-8], [66], [-20], [6], [30], [20], [24], [16], [-52], [-62], [-76], [-26], [-32], [-60], [-16], [-22], [48], [-24], [68], [-54], [-56], [-68], [-6], [-70], [-30], [4], [-54], [-36], [-60], [78], [-12], [80], [-24], [-62], [-60], [-84], [-44], [-50], [6], [-64], [-60], [20], [-58], [60], [16], [-18], [-34], [6], [-60], [90], [-4], [-66], [-74], [-12], [48], [-50], [-30], [-32], [50], [18], [-46], [-42], [10], [-16], [28], [24], [12], [-22], [-42], [-8], [-48], [92], [12], [-76], [-30], [-38], [88], [-28], [-42], [42], [-46], [6], [-60], [-74], [-64], [60], [72], [64], [38], [36], [46], [-4], [54], [42], [12], [30], [24], [26], [6], [84], [-12], [-92], [26], [0], [74], [-34], [-84], [28], [66], [-90], [4], [38], [-36], [66], [-38], [-54], [34], [18], [102], [-52], [-30], [4], [36], [8], [-14], [20], [-12], [-22], [-78], [-36], [52], [-24], [86], [22], [-6], [-64], [24], [64], [-90], [50], [-90], [-18], [-8], [62], [-78], [4], [-60], [22], [-24], [12], [-4], [-22], [-72], [-84], [10], [42], [-12], [-42], [-44], [-90], [-60], [-14], [54], [-12], [-6], [52], [-18]];
    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_u();".
// 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_u(: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_u();".
// 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_u( : 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![-4, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;