
// Make newform 5544.2.a.f in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5544_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_5544_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_5544_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_5544_a();"
function MakeCharacter_5544_a()
    N := 5544;
    order := 1;
    char_gens := [4159, 2773, 4313, 1585, 2521];
    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_5544_a_Hecke(Kf)
    return MakeCharacter_5544_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], [0], [-2], [1], [1], [4], [6], [2], [2], [2], [8], [-2], [-6], [-2], [-10], [-14], [0], [12], [-8], [6], [6], [4], [4], [-12], [14], [-10], [-16], [4], [16], [20], [20], [20], [0], [-6], [-10], [20], [14], [-16], [-8], [-6], [4], [-18], [-6], [14], [14], [24], [-6], [-16], [12], [22], [6], [28], [-22], [-12], [-28], [16], [-6], [28], [-12], [-10], [18], [-6], [2], [22], [-26], [-22], [8], [-2], [20], [0], [4], [-8], [16], [36], [-24], [6], [-26], [-34], [16], [-14], [-20], [-10], [24], [-2], [-8], [24], [12], [10], [-30], [8], [-12], [12], [40], [12], [4], [0], [-6], [-36], [-14], [32], [34], [26], [-4], [-6], [-22], [22], [-16], [-6], [-30], [18], [-20], [32], [28], [8], [8], [28], [16], [30], [14], [4], [6], [-26], [6], [-28], [-24], [-14], [-38], [42], [-32], [-28], [6], [48], [48], [-10], [6], [22], [-14], [2], [22], [10], [-2], [-50], [-24], [-52], [-22], [42], [-12], [6], [-12], [6], [12], [48], [-24], [-28], [-4], [-34], [36], [20], [-14], [46], [48], [-14], [-28], [48], [32], [14], [24], [-8], [-30], [14], [-52], [-50], [8], [-26], [24], [-8], [-54], [22], [16], [-16], [-56], [28], [-2], [-34], [-2], [6], [60], [-56], [-14], [60], [-6], [12], [-12], [6], [-28], [-44], [58], [-46], [-42], [0], [14], [32], [22], [-10], [36], [-30], [48], [-60], [2], [20], [38], [22], [0], [60], [36], [-30], [-16], [-30], [-10], [30], [52], [-28], [-24], [-56], [-28], [4], [-56], [54], [-64], [20], [-74], [-2], [-56], [10], [-56], [-24], [-50], [-30], [-12], [-30], [-68], [-22], [-16], [34], [58], [20], [-24], [-48], [42], [48], [-12], [10], [34], [-58], [-74], [12], [54], [38], [-22], [-26], [52], [-76], [-28], [8], [48], [-60], [-42], [24], [14], [-6], [-58], [60], [-46], [-24], [-2], [-64], [-48], [-40], [-46], [-20], [-52], [-32], [28], [0], [22], [-54], [-26], [-6], [56], [-70], [-34], [24], [78], [-4], [-4], [-66], [40], [10], [36], [-58], [6], [-22], [64], [-20], [24], [-14], [8], [26], [-26], [4], [-16], [66], [-18], [-28], [-48], [-42], [-4], [-68], [30], [-22], [10], [-90], [-42], [-40], [54], [50], [64], [8], [84], [-38], [76], [-74], [0], [60], [-2], [-68], [-38], [-54], [6], [-12], [-54], [68], [26], [24], [42], [-4], [22], [80], [-84], [66], [-34], [70], [66], [-36], [80], [94], [-14], [-76], [2], [50], [48], [42], [58], [-4], [28], [-38], [-10], [-60], [-18], [40], [-28], [68], [30], [80], [-26], [64], [60], [-86], [-90], [-14], [54], [28], [44], [34], [-22], [-42], [-16], [98], [-58], [-42], [-34], [-58], [-36], [-16], [-34], [26], [0], [56], [64], [-54], [-60], [-32], [-48], [-80], [-70], [-56], [78], [62], [-60], [-56], [-6], [22], [-96], [26], [-18], [50], [-60], [-32], [24], [88], [14], [36], [30], [-52], [70], [-30], [-16], [86], [68], [-76]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5544_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5544_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_5544_2_a_f(:prec:=1)
    chi := MakeCharacter_5544_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_5544_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_5544_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5544_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<13,R![-4, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;