
// Make newform 5544.2.a.q in Magma, downloaded from the LMFDB on 08 June 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_q();".
// 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_q();".
// 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]];
    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], [1], [1], [1], [-5], [6], [-1], [-4], [-1], [-10], [1], [0], [-8], [-1], [-8], [3], [-6], [13], [0], [-3], [-8], [-2], [6], [-16], [-10], [2], [7], [10], [14], [-22], [8], [-6], [0], [-13], [-10], [20], [-19], [-8], [0], [4], [12], [-18], [20], [14], [0], [-18], [-22], [6], [16], [-18], [13], [-1], [-3], [17], [-5], [18], [19], [12], [-13], [-9], [0], [-4], [-32], [-14], [8], [8], [28], [-16], [15], [-5], [4], [-26], [-18], [-21], [-24], [-20], [2], [28], [-14], [-35], [-1], [-21], [28], [-5], [-24], [-24], [-32], [24], [29], [-33], [42], [-32], [15], [-5], [24], [-30], [-39], [1], [20], [28], [-37], [32], [42], [-4], [-2], [41], [36], [-24], [-15], [19], [2], [22], [-46], [20], [-38], [-8], [21], [-16], [19], [36], [-8], [42], [-22], [-6], [22], [-5], [-15], [-8], [2], [0], [51], [21], [17], [24], [-41], [-11], [-7], [19], [31], [25], [-11], [-3], [23], [-16], [-9], [-42], [-30], [24], [36], [0], [33], [3], [14], [-52], [26], [-60], [59], [22], [28], [-42], [-17], [-46], [9], [8], [32], [21], [46], [18], [41], [23], [-50], [-40], [-14], [-30], [58], [36], [-38], [-23], [41], [-11], [-8], [-38], [8], [4], [-15], [12], [-2], [-14], [-66], [42], [57], [45], [3], [23], [-29], [46], [2], [48], [60], [50], [-7], [40], [-55], [60], [66], [-24], [21], [-70], [32], [-58], [-17], [-57], [48], [30], [42], [68], [-54], [2], [18], [-26], [41], [-60], [31], [8], [-26], [43], [-54], [-31], [-19], [-26], [37], [-38], [-17], [76], [54], [-14], [39], [30], [-45], [16], [44], [-34], [34], [-44], [20], [-12], [3], [-48], [-21], [-42], [-8], [-62], [-46], [-50], [-36], [-3], [17], [-49], [13], [-50], [20], [62], [65], [63], [18], [-6], [9], [-4], [-30], [32], [-51], [44], [-54], [25], [-22], [0], [-10], [50], [-8], [-16], [40], [22], [45], [-5], [36], [19], [42], [-82], [-46], [-58], [-24], [-33], [44], [32], [-84], [-26], [34], [0], [23], [63], [-55], [16], [-8], [81], [19], [44], [-16], [-68], [-56], [-34], [-36], [15], [-76], [48], [-6], [20], [40], [-12], [-70], [55], [-9], [42], [32], [-15], [8], [4], [-64], [-15], [43], [-80], [76], [-9], [-18], [4], [-41], [7], [-12], [54], [48], [-6], [14], [-10], [15], [-66], [71], [-74], [-64], [0], [-90], [20], [-83], [36], [-9], [74], [-38], [-5], [-76], [23], [-4], [-48], [-84], [-56], [95], [40], [-17], [-58], [42], [66], [82], [-82], [-7], [-39], [-40], [-17], [-71], [21], [61], [66], [22], [-21], [-68], [-34], [-65], [-4], [15], [-19], [62], [-58], [-27], [5], [38], [12], [-64], [38], [-58], [-57], [-25], [-56], [3], [-9], [-98], [66], [-14], [-28], [-71], [-81], [-22], [-6], [64], [54], [106], [48], [56], [-6], [-67], [-33], [-74], [30], [22], [86], [-93], [78], [-10], [-14], [81], [-22], [-37], [-10], [20]];
    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_q();".
// 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_q(: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_q();".
// 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_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5544_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1, 1]>,<13,R![5, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;