
// Make newform 5586.2.a.i in Magma, downloaded from the LMFDB on 19 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_i();".
// 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_i();".
// 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], [-4], [2], [2], [1], [-4], [2], [-4], [-6], [10], [-4], [8], [10], [-12], [2], [-8], [-16], [-10], [-8], [0], [10], [2], [-6], [4], [12], [2], [-10], [0], [0], [-14], [4], [-6], [16], [-14], [4], [-24], [6], [12], [-22], [12], [-22], [18], [-8], [16], [12], [-12], [-6], [2], [20], [10], [-16], [18], [-12], [6], [-8], [-10], [6], [-4], [30], [28], [-24], [-10], [18], [-16], [-30], [-12], [-14], [26], [-4], [-8], [-30], [-32], [8], [-30], [2], [-10], [10], [-24], [-22], [16], [26], [-20], [12], [14], [-6], [-6], [-40], [0], [-16], [-16], [-20], [-20], [-16], [-10], [-22], [-4], [-10], [-32], [2], [36], [-42], [4], [-34], [16], [-6], [0], [-22], [-4], [14], [-6], [20], [40], [-42], [44], [-48], [-6], [-12], [26], [34], [22], [-4], [-52], [18], [-26], [16], [24], [18], [44], [0], [32], [-2], [-30], [-34], [30], [-4], [-10], [2], [20], [-46], [-32], [36], [26], [48], [-22], [26], [-44], [-40], [-46], [10], [-20], [-32], [8], [8], [32], [58], [30], [6], [-12], [54], [32], [12], [6], [48], [40], [-6], [34], [18], [-20], [-22], [28], [10], [60], [-46], [20], [2], [-36], [-46], [32], [-20], [58], [42], [-44], [50], [58], [4], [-62], [0], [2], [-20], [32], [-18], [-36], [-6], [-14], [38], [-14], [-40], [18], [44], [18], [-46], [-40], [34], [64], [52], [50], [-52], [18], [50], [0], [60], [0], [-22], [16], [18], [24], [-54], [-54], [36], [-42], [-56], [-36], [38], [26], [-56], [28], [36], [54], [20], [40], [-2], [68], [64], [-50], [-22], [-12], [-16], [-36], [20], [-48], [34], [18], [16], [40], [12], [-40], [-20], [54], [10], [36], [-54], [-22], [-12], [70], [-52], [22], [62], [56], [-44], [10], [50], [18], [-76], [-14], [-46], [-48], [34], [-54], [-24], [54], [-72], [-38], [-56], [60], [-14], [42], [72], [-32], [-72], [-48], [-22], [44], [-84], [-78], [-22], [-64], [50], [18], [0], [-10], [-36], [-14], [34], [28], [-6], [-36], [36], [-82], [50], [-72], [44], [68], [58], [4], [58], [-44], [-74], [-56], [10], [-74], [8], [36], [82], [-72], [80], [70], [82], [-36], [10], [30], [-64], [-6], [10], [24], [68], [-8], [-30], [22], [2], [-8], [24], [4], [72], [34], [-62], [-2], [-32], [82], [18], [70], [40], [34], [68], [26], [-16], [-48], [58], [-4], [-22], [-70], [-12], [50], [-54], [16], [-16], [2], [64], [62], [10], [-48], [60], [20], [-78], [42], [-24], [-78], [-92], [68], [44], [74], [32], [62], [84], [-92], [62], [50], [2], [-82], [66], [72], [98], [-44], [-24], [-8], [42], [84], [96], [-46], [-42], [44], [-12], [-8], [-62], [-52], [42], [88], [-14], [-46], [74], [64], [-102], [-82], [-40], [-18], [-50], [68], [-56], [18], [-30], [84], [-100], [-38], [-38], [-80], [-12], [-62], [24], [-30], [-22], [84], [-12], [34], [70], [-12], [50], [-100], [-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_i();".
// 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_i(: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_i();".
// 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_i( : 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![4, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>],Snew);
    return Vf;
end function;