
// Make newform 5520.2.a.bs in Magma, downloaded from the LMFDB on 25 April 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_bs();".
// 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_bs();".
// 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 := [-4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 2]];
    Rf_basisdens := [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], [1, 0], [1, 0], [1, 1], [-1, -1], [2, 0], [3, 1], [-4, 0], [-1, 0], [2, 0], [0, 0], [-3, -1], [2, 0], [0, 0], [8, 0], [4, -2], [6, 2], [5, -1], [8, 0], [2, -2], [4, 2], [1, 1], [1, -3], [-1, 1], [-8, 2], [-8, -2], [-3, -3], [1, -3], [7, 1], [-5, 1], [14, 2], [4, 0], [-15, -1], [0, 4], [-10, 0], [2, 2], [5, -1], [12, 0], [8, 0], [12, -2], [-8, -4], [15, 1], [4, 4], [0, -2], [-10, 0], [-15, 1], [0, -4], [2, -2], [-7, -3], [11, -3], [4, -6], [10, -2], [14, -4], [-5, 3], [0, -2], [-2, -2], [-8, -2], [-2, -2], [12, 2], [-17, 1], [-6, 2], [20, -2], [16, -4], [-2, -6], [-6, 4], [-4, 6], [16, 4], [10, -4], [10, 6], [0, -6], [-14, 0], [-4, 4], [-9, -1], [1, 3], [-4, 0], [-22, 2], [-18, 0], [-18, -4], [-5, -3], [-8, -2], [-19, 5], [1, 3], [-18, -2], [-6, 4], [0, 0], [8, 4], [6, -4], [-8, -6], [8, 6], [36, 0], [-9, 3], [4, 4], [-24, -4], [12, 0], [-4, 8], [-18, 6], [34, 0], [13, -1], [12, 4], [-24, 2], [16, -4], [-2, 0], [1, -3], [23, 1], [-10, -6], [-2, 4], [18, -2], [16, -2], [-2, 2], [-6, 0], [-2, -6], [39, 1], [3, 9], [36, 0], [27, 3], [3, -3], [-20, -4], [-28, 4], [-6, 4], [-5, 3], [5, 7], [-14, -8], [-30, -4], [-26, 2], [-20, 0], [-28, -2], [1, 3], [18, -2], [1, -7], [15, -7], [-24, -4], [-10, 6], [-15, -7], [-19, 7], [-40, -2], [4, -6], [-28, -2], [-48, 0], [-6, -4], [-4, 2], [36, 0], [-28, 2], [4, 0], [-13, -1], [24, 2], [18, 2], [-6, 0], [0, 6], [-32, -4], [-16, -8], [28, -6], [11, 5], [-12, 8], [-36, -4], [-46, 2], [22, -2], [-15, 1], [2, -8], [8, -6], [10, -4], [-14, -2], [-5, -7], [-28, 8], [-39, 1], [-7, 7], [18, 2], [-12, 12], [12, 2], [-26, 4], [-26, 0], [7, -1], [16, 2], [-2, 6], [-10, 8], [-10, -2], [-3, -9], [-12, 0], [18, 0], [-43, -3], [-41, 1], [-8, -12], [-9, 7], [28, 2], [-42, -4], [10, 10], [10, -4], [26, -8], [32, -8], [8, -2], [-18, 10], [8, 6], [38, -6], [-42, 2], [-26, 4], [17, -3], [17, -1], [-22, 0], [-9, 1], [29, 3], [32, 0], [2, -4], [58, 2], [-8, 6], [-20, -6], [-57, -1], [0, 2], [3, 3], [-36, 8], [-18, 4], [-36, 0], [-20, 2], [-2, 4], [-13, -13], [31, 3], [10, 6], [-16, -2], [-38, -2], [8, 6], [-28, 4], [-20, -10], [14, -8], [13, -3], [52, 2], [41, -7], [-32, -4], [-46, -4], [-33, -3], [42, 6], [7, -9], [-18, -6], [40, -2], [-44, 0], [-7, 1], [20, 2], [-26, -2], [-14, -6], [-30, 8], [-6, 4], [12, -8], [2, -10], [-19, 9], [-4, -8], [4, -16], [2, -4], [8, -10], [22, -6], [18, 6], [-47, -7], [-20, 8], [52, -4], [9, 11], [5, -1], [6, -2], [16, -2], [-60, -2], [-38, 6], [47, 1], [20, -4], [18, -12], [-16, 6], [41, 1], [33, 5], [-12, 14], [-5, -3], [40, -10], [44, -8], [-18, 8], [61, -1], [2, 2], [-6, 12], [44, -2], [8, -8], [42, -4], [29, -3], [-2, 4], [-16, -4], [28, 0], [-6, -12], [-22, 8], [-25, 7], [32, 0], [-17, 7], [0, 0], [-11, -9], [-12, -16], [-30, 6], [32, 2], [-22, 4], [16, 0], [56, -2], [-14, 4], [1, 5], [-14, 8], [-31, 9], [20, 10], [6, 8], [-29, 11], [34, 12], [50, -2], [0, -4], [-34, -8], [30, 0], [-9, -17], [28, 8], [16, -4], [-12, 10], [-16, -4], [-29, 13], [30, 14], [54, 4], [-4, -12], [-36, 6], [45, 7], [-8, -12], [2, 2], [-16, 6], [14, 10], [10, 6], [58, 4], [18, -16], [10, -2], [-22, -12], [-10, -4], [6, 10], [11, 1], [-2, -4], [-10, 10], [0, 4], [-4, 12], [26, -12], [-4, -2], [-14, -4], [-52, 4], [-42, 2], [-20, -8], [-14, -2], [39, 1], [-37, -7], [0, -10], [1, 17], [78, -4], [-13, -7], [42, 0], [-13, -5], [-24, 10], [62, 2], [14, 8], [64, -4], [-38, 2], [-60, -2], [-4, -16], [26, 0], [-28, 2], [31, 15], [-27, 7], [-22, 0], [70, -2], [-5, -5], [2, 0], [56, 0], [31, -7], [56, 6], [-44, -12], [-55, 1], [72, 4], [40, 6], [-4, 6], [-41, -9], [-68, -6], [48, -4], [-48, 12], [0, 0], [16, -14], [43, 11], [26, 16], [-12, 0], [54, -2], [-20, -2], [1, -5], [4, 18], [50, -4], [-35, 11], [-6, -2], [80, -2], [-40, 4], [-64, 0], [30, 6], [-28, -6], [36, 4], [-42, -2], [12, 2], [26, 12], [54, 10], [24, 4], [-6, 2], [-36, 6], [-11, -19], [21, 15], [-22, -2], [14, 4], [22, 8], [18, 0], [3, 3], [-25, -11], [-54, 0], [-42, 6], [57, -5], [-4, 10], [10, 10], [-8, 12], [34, -16], [0, 10], [-19, -15], [6, -6], [6, 0], [-2, -12], [-30, -10], [-24, 20], [45, 3], [-8, 0], [50, 12], [-35, -1], [16, 0], [18, -2], [-6, -16], [50, -12], [29, -15], [-22, 0], [20, -8], [-6, -2]];
    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_bs();".
// 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_bs(:prec:=2)
    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_bs();".
// 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_bs( : 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, -2, 1]>,<11,R![-16, 2, 1]>,<13,R![-2, 1]>,<17,R![-8, -6, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;