
// Make newform 5520.2.a.ca in Magma, downloaded from the LMFDB on 08 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5520_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_5520_2_a_ca();".
// 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_5520_2_a_ca();".
// 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
        poly := [-1, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-3, 1, 1], [5, 2, -2]];
    Rf_basisdens := [1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_5520_a();"
function MakeCharacter_5520_a()
    N := 5520;
    order := 1;
    char_gens := [4831, 1381, 1841, 4417, 1201];
    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_5520_a_Hecke(Kf)
    return MakeCharacter_5520_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, 0], [-1, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 0], [1, 0, -1], [2, -1, 0], [0, 0, 0], [-1, 0, 0], [2, 1, 0], [-1, -1, -1], [4, 1, 2], [1, 1, -1], [-1, -2, -1], [0, 2, 0], [5, 1, -1], [-1, 1, -1], [-1, 2, 1], [2, 1, 2], [-2, 1, -2], [2, -2, 0], [-3, -2, 1], [-2, -1, 2], [8, 2, 2], [1, -2, -1], [2, 1, 0], [1, -2, 1], [4, -1, 0], [3, 0, -3], [4, -1, 2], [3, 0, -1], [2, 0, 2], [1, -2, -1], [3, -3, -1], [6, 0, 0], [0, 0, 0], [2, 3, 0], [10, 2, -2], [8, 2, 0], [10, 2, 0], [0, 0, 0], [-1, -4, 1], [2, -2, 2], [-4, -2, 2], [4, -4, 2], [-3, 0, 1], [3, 3, -1], [9, 0, -3], [3, -6, -1], [11, 4, 5], [4, -2, -2], [-6, 7, 2], [-8, -2, -2], [-4, -6, -4], [6, -4, 0], [7, -1, -1], [18, 3, 0], [11, 1, 3], [1, 6, -1], [8, 6, 2], [12, 3, 4], [7, -5, 1], [6, 0, 2], [-1, -6, -1], [2, 3, 0], [6, 4, -4], [3, -5, -1], [-7, 6, -1], [12, 0, 0], [-1, -3, 1], [4, 4, -2], [-14, -2, 2], [0, 3, 0], [-3, -2, -5], [14, -4, -2], [7, -1, -1], [12, 0, -2], [-3, 0, 3], [-4, -4, 6], [5, -1, 3], [-2, 4, -2], [-7, -4, 3], [2, 4, 2], [-2, 5, 4], [6, 4, 6], [20, -2, 0], [15, -1, -3], [-2, -1, -4], [-13, 6, 1], [21, -2, 1], [16, -1, -4], [-10, 6, -2], [-9, -4, -5], [5, -1, -3], [3, -3, 7], [9, -1, 1], [-9, -6, -3], [6, 2, 0], [17, 2, 1], [2, -8, -4], [0, 0, -4], [-1, -3, 1], [28, 1, 0], [28, -4, -2], [6, 10, 6], [-14, 6, 0], [14, -6, 2], [6, 0, 0], [-21, -2, -5], [-29, 5, 1], [5, 0, 1], [-11, -6, -5], [-16, 5, -2], [8, -4, 0], [-7, 0, 5], [-4, -6, -2], [12, 5, 4], [4, 0, -4], [-8, 6, 6], [8, -8, 0], [11, 6, 5], [-28, -4, -6], [-23, -1, -5], [26, -6, -2], [-18, -8, -6], [-6, -4, 4], [7, 2, 1], [-18, -5, 6], [-20, -1, 4], [-10, 1, 4], [21, -5, 1], [12, 0, 4], [-17, 2, -5], [-12, -1, -6], [-15, 1, -1], [-10, -8, 4], [20, 4, -2], [-8, 13, 0], [-31, -1, 3], [27, 3, 1], [9, -9, -3], [1, -2, 7], [-1, -2, -5], [4, 9, 0], [-3, -13, -1], [-14, -10, 2], [-9, 10, 5], [-22, 0, 4], [-3, 3, -7], [6, 2, -2], [-7, 2, 7], [6, 8, -8], [12, -2, 0], [24, 6, 8], [-12, -3, 4], [-32, -4, 0], [-29, -6, -1], [-1, 3, -3], [9, -2, 7], [-22, -4, -4], [-22, -4, 6], [7, 2, -3], [-37, 4, -1], [-12, -8, 4], [-22, -7, 0], [17, 9, -7], [-7, 7, 1], [11, -8, 1], [0, 10, 6], [-4, 4, 2], [22, 0, -2], [-29, -1, -3], [-20, 6, -4], [-18, 9, 4], [-20, 0, 4], [30, -4, 0], [-6, -4, -2], [-4, -11, 2], [-4, 3, -4], [-1, -8, -7], [-3, 14, 1], [-8, 14, 0], [21, -6, 3], [18, 6, -4], [5, -5, 5], [2, 10, 4], [-15, -8, -1], [-8, 3, 0], [-19, -1, -5], [-26, -3, 6], [-24, -10, -2], [32, 2, -4], [-20, 4, 4], [-12, -1, 2], [-7, 10, 5], [-8, -3, -10], [-2, 6, -4], [-5, 6, -7], [-7, -6, 7], [-52, -2, -4], [-20, 0, 6], [-28, 0, -4], [25, 0, -1], [2, -2, 8], [-12, 6, 4], [-20, 16, 2], [-3, 0, -7], [12, 0, 8], [7, -11, -3], [-14, -8, -10], [8, -20, -2], [-9, 6, 5], [-25, 10, -1], [11, 2, -9], [-48, -5, 0], [-22, -4, 0], [3, 4, 7], [41, 1, -1], [1, -13, -7], [16, 6, -2], [-23, 3, -5], [-11, -8, 1], [7, 3, 5], [-16, -9, 8], [-12, 16, 0], [-5, -7, -3], [39, 2, 5], [-16, 3, -8], [-13, 6, 3], [13, 3, 5], [41, 8, 7], [2, -16, 2], [-27, 0, -7], [1, 11, -1], [-2, -3, 6], [0, -12, 0], [20, 0, -6], [14, -16, 0], [-23, -3, 1], [-18, 13, 6], [-18, -5, 2], [3, -13, -1], [-19, -12, 9], [12, 12, 6], [24, -6, -6], [7, -6, -1], [15, 12, 11], [22, 12, -2], [18, 10, 2], [-13, 1, 11], [-5, 14, 1], [32, -12, 2], [9, -13, 1], [30, -4, -4], [8, -6, 6], [-11, 1, -3], [49, 6, 3], [-35, -10, -3], [28, 6, -6], [10, 0, 8], [2, -17, 2], [14, -3, -6], [1, 5, -13], [38, -5, -4], [-8, -2, 2], [20, 14, -4], [4, -4, -2], [4, -24, -2], [-18, -1, -10], [-20, -10, -6], [-5, -9, 5], [-20, -15, -4], [22, 5, -4], [3, 12, -1], [-20, 4, 2], [-9, 2, 3], [-18, -14, -14], [-13, 3, 5], [-20, -16, 2], [16, -10, 0], [0, -14, -8], [17, 8, 5], [46, 0, 6], [-15, -12, 3], [-16, -4, -12], [-14, -7, 2], [-22, -15, 0], [2, 0, 4], [2, 20, 2], [-18, -12, 4], [-50, -4, 0], [-18, -7, 2], [46, 2, 0], [10, 2, 2], [-33, -10, 5], [-28, 6, 6], [-19, 6, -7], [22, -2, -4], [17, 15, 9], [-8, -18, -4], [38, 3, 4], [-21, -11, -3], [25, 14, -3], [-2, -10, 2], [4, 0, -4], [16, 12, -2], [22, 16, 10], [-19, -2, -1], [6, 6, -2], [-3, 22, 3], [4, -12, -4], [42, -2, 4], [0, -4, -14], [6, 12, -6], [31, -1, 7], [44, 8, 10], [9, -7, -7], [28, -5, -4], [-46, -3, 0], [-42, -8, 4], [-28, -14, 4], [-18, -13, -4], [-43, 2, 3], [-21, 8, 7], [38, 3, -4], [-14, 8, -8], [4, -2, 12], [2, 0, 14], [28, 16, 4], [21, -1, -1], [-7, 9, -13], [-40, -10, -10], [19, 13, -5], [14, -8, -6], [-12, 8, -4], [32, -4, -4], [-15, 2, 11], [-19, -6, 11], [26, 0, 0], [-6, 5, -6], [-7, -14, -1], [1, -2, 15], [-73, 2, -3], [-11, 12, 1], [-1, 7, -7], [-15, -21, 1], [57, -5, -5], [-68, 8, 0], [-18, -18, 6], [35, -5, -3], [11, 13, -1], [-10, -8, -4], [-6, 25, 8], [-26, -17, -10], [-13, -2, -3], [12, 4, 14], [12, 0, 4], [-2, -2, 6], [-10, 22, 8], [-42, 6, 6], [-36, -11, 2], [7, 1, 13], [32, -10, -8], [-36, 0, 4], [12, 6, -8], [24, 8, -2], [-34, -4, -4], [-26, 9, -2], [20, -6, 10], [1, -5, -7], [59, 7, -1], [20, -22, -4], [-16, -6, -14], [-19, 18, 1], [7, 4, 13], [31, -15, -1], [-6, -10, 10], [46, 1, -4], [-66, 2, 0], [24, 0, -2], [32, -16, -6], [9, 6, 7], [7, 10, -5], [-44, -2, 6], [62, 8, 6], [-12, -10, -4], [19, -1, 3], [-23, 2, 7], [-30, 17, 2], [61, -3, 5], [-36, -2, 2], [-2, -12, 12], [-21, 11, 11], [-6, -14, 14], [50, 4, -6], [14, 1, -4], [-7, 0, 13], [-32, -6, -14], [-8, -4, 16], [32, -1, 6], [39, -11, 1], [36, 8, 6], [82, -1, 2], [-18, -3, -4], [10, -15, 8], [29, -3, -3], [28, -25, -6], [32, 0, -2], [-1, 10, -9], [32, 12, 8], [-22, 4, -8], [-38, -6, 0], [-23, 14, -3], [-46, -4, -6], [30, -9, 4], [-27, 2, -5], [-47, 2, 1], [-11, 16, 9], [38, 3, 4], [-20, 0, 12], [-14, -10, 12], [37, -6, 3], [36, 4, -4], [22, 8, -10], [6, -12, -12], [-42, -18, 4], [-49, -10, 3], [-39, -9, -9], [20, 16, -8], [-10, 5, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5520_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5520_2_a_ca();".
// 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_5520_2_a_ca(:prec:=3)
    chi := MakeCharacter_5520_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_5520_2_a_ca();".
// 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_5520_2_a_ca( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-16, -12, 1, 1]>,<11,R![0, 1]>,<13,R![16, -20, -4, 1]>,<17,R![28, 4, -7, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;