
// Make newform 5520.2.a.cb 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_cb();".
// 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_cb();".
// 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 := [2, 3, -9, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-2, -11, 0, 1], [-8, 9, 2, -1], [2, -8, -1, 1]];
    Rf_basisdens := [1, 2, 2, 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, 0], [1, 0, 0, 0], [-1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [-1, 1, -1, 0], [1, 0, 1, 2], [1, -1, -1, 0], [-1, 0, 0, 0], [0, 1, -2, -1], [1, 0, -1, -1], [0, 1, 2, 0], [2, -1, 0, 1], [6, 0, 2, 0], [-1, 1, 1, 0], [0, 1, 2, -1], [2, -1, 2, 1], [0, 0, 2, 1], [7, 2, 1, 1], [0, 1, 0, -1], [-5, 1, -1, 0], [3, -1, -3, -1], [7, 0, 1, -2], [3, -1, -1, -1], [-4, 0, -2, -2], [4, -1, 2, -1], [9, -1, -1, 1], [1, 0, -1, 0], [2, 0, 0, 1], [-7, -2, 1, 0], [7, 3, 1, 2], [4, 4, 0, 0], [1, 1, 1, 3], [9, 0, 3, -1], [-1, -3, 3, 2], [6, -4, 2, 0], [-4, 1, -2, -4], [14, 0, -2, -2], [1, -5, -1, 0], [-10, 2, 0, 0], [0, -2, -4, -2], [1, 1, 1, 5], [-3, -1, 3, 4], [-4, 0, -2, -4], [8, 2, -2, -4], [3, 1, 5, 1], [11, -4, 1, 5], [6, 4, 2, 2], [7, -1, 1, 3], [1, -3, 1, 5], [-10, -2, -4, -4], [-12, -1, 4, 5], [-3, -5, 5, 2], [1, 3, -1, 1], [1, 3, 1, 0], [0, -3, 0, -3], [4, 3, 2, -1], [5, 4, 3, 1], [6, -4, 0, 0], [-6, 6, 0, -3], [-3, 4, -5, -1], [-2, 3, 4, -1], [7, 1, -3, -8], [4, 0, -4, -8], [-5, -2, 7, 3], [13, -1, 1, 4], [3, -2, 1, 3], [4, -2, -2, -2], [-4, 0, 0, -2], [9, -2, -3, -3], [-9, 1, -5, -6], [-9, 5, 1, -4], [18, -1, 6, -2], [-17, -1, 3, 3], [-4, 0, 4, 4], [-8, 3, 0, -5], [0, -2, 2, 8], [2, -2, 4, 4], [-3, -5, 5, 1], [15, -2, 3, -1], [0, 8, 0, -1], [8, -2, -6, -1], [-20, 4, -4, -2], [-17, 2, -5, -1], [2, -4, -2, 6], [-3, -1, 7, 4], [-2, 7, -4, 1], [17, 4, 1, 1], [8, 2, 6, 0], [-1, -7, -7, -2], [-9, -2, 1, -2], [-7, 3, -1, 4], [1, -3, -1, -4], [-14, 3, -6, -1], [5, -2, 7, 1], [0, -5, 8, 3], [-10, -2, 0, -4], [-2, 2, 4, 3], [-10, -6, -6, -2], [-2, 2, 8, 0], [16, 0, -4, 0], [6, -1, -4, -3], [13, -2, 3, 2], [1, 3, -7, 1], [-5, 3, -3, -8], [-8, 8, -6, 2], [8, -6, -4, 4], [-9, 5, -5, 0], [-6, 6, -2, 0], [13, -4, -3, -3], [-15, 5, -1, -2], [-7, -1, 1, -5], [-13, 0, -1, -8], [-8, -8, 0, -4], [-10, 2, 2, 7], [12, -8, -2, 5], [5, -4, -5, 1], [-9, -1, 1, -2], [9, -7, 5, 8], [-24, 0, -8, 5], [-3, 3, -11, -3], [7, -5, 3, 0], [2, -3, 0, -9], [-11, -5, -9, -2], [24, 2, 4, -2], [7, -7, -5, 0], [4, 4, -2, 5], [-14, -1, -2, -5], [20, 3, -4, -2], [-4, -7, -2, 6], [-1, 2, 5, -1], [-10, -6, 2, -6], [-4, 6, 4, -1], [-18, -1, -4, 0], [-14, -3, 0, 1], [13, 1, 5, 6], [6, -4, -4, -2], [25, 4, 7, 1], [-6, 1, -8, 3], [16, -1, 2, -3], [7, -8, -3, 5], [4, 0, -6, -10], [-10, -6, -6, 2], [13, 2, 3, 4], [-9, 4, -1, -7], [4, 2, 4, 0], [10, -8, 4, -2], [12, -2, 6, -6], [-17, -4, -3, 5], [8, -2, 0, 6], [12, -10, 2, 8], [0, -2, 2, 3], [-25, 3, -3, 2], [-4, 6, 4, 6], [21, -4, 3, 3], [14, -6, -6, 2], [11, -1, -3, 3], [8, 1, 2, 9], [0, 8, -6, 2], [-7, -5, 1, 6], [12, 4, -8, -10], [-11, -5, 13, 7], [-3, 5, -5, 0], [15, -5, 9, 1], [-25, 0, -5, 2], [-6, 3, -2, 1], [3, 12, -3, 1], [-10, -10, 0, 2], [28, 0, -2, 2], [-17, -7, -9, -6], [-29, -1, -3, -9], [21, -4, 1, -5], [36, 2, 4, 0], [5, -12, 5, 3], [6, -2, -14, -10], [-22, 0, 0, -3], [2, 12, -6, 2], [6, 7, 8, 7], [-2, -5, 10, 0], [-12, 2, -10, -3], [13, -1, 3, 2], [6, 4, -6, -5], [4, -2, 2, -8], [-5, 3, -1, 8], [-16, 5, 0, -3], [-23, 9, 1, -4], [3, 1, -5, -16], [-15, 10, -1, -1], [11, 4, -9, -11], [12, -7, 4, 3], [-1, 5, 3, 0], [-8, 2, -4, -12], [5, -9, 3, 2], [-18, 3, -8, -5], [-1, 9, 1, -3], [-13, 6, -1, -12], [-31, -5, -7, -2], [-13, -5, 7, 1], [-1, -3, 3, -3], [6, 0, 2, 14], [-21, -3, -9, -2], [18, 4, 6, 0], [3, -3, -5, -10], [25, 7, 1, 0], [-16, -6, 0, -5], [-18, -8, -8, -2], [20, 0, 4, 11], [7, -1, -3, 6], [-34, -5, 4, 1], [8, -6, -4, -10], [-2, 8, -4, 2], [14, 8, -8, -10], [-33, -7, -7, -1], [29, -1, 3, -1], [6, -5, 10, 7], [-19, -3, -3, 6], [-20, 10, -4, -14], [-8, -7, 2, -1], [-22, -13, -10, -5], [-3, 1, 9, 10], [-7, 14, -3, -1], [7, -3, 9, 5], [-14, -1, 8, 3], [-30, -13, -2, -8], [11, -5, 1, 14], [47, 4, 7, -1], [-29, 5, -9, -5], [12, 7, 4, 9], [-27, -3, -13, 1], [20, 5, -8, -3], [-39, -1, -3, -6], [-13, 9, -11, -8], [-9, 1, 1, -3], [16, 1, 2, 11], [-21, 6, -3, -5], [-14, 4, 6, 8], [29, -11, 5, 0], [38, 0, -4, -8], [14, -1, 6, -1], [6, 1, 2, 13], [9, -6, 7, 6], [-15, -4, -13, 3], [-18, 12, 2, 0], [-44, 0, 10, 6], [-26, -10, 4, 0], [10, -2, -2, -8], [-5, -9, 13, 14], [4, 8, -4, -1], [-32, -14, -8, -6], [-2, 3, 10, 7], [-19, 9, 5, -1], [-27, -1, 5, 3], [-4, -9, 4, -5], [22, 0, 4, -2], [-19, -7, -7, -6], [-10, -9, 14, 9], [6, -2, 4, 15], [14, -2, 2, 2], [11, 11, 3, 4], [24, -6, 10, -2], [-24, -7, 8, 8], [15, 4, 9, -2], [9, -12, 5, 11], [-20, 5, 6, -6], [20, 4, 6, 4], [13, 1, 11, 6], [21, -1, -3, -10], [9, -17, 9, -1], [-39, -4, -1, -1], [-8, 4, 14, 2], [13, -14, -7, 9], [41, -8, 7, -3], [-27, -6, 5, 1], [5, -13, 3, 5], [1, -7, -7, -4], [-7, 3, -9, -22], [-19, -9, -1, 8], [7, 0, -9, -5], [29, 9, 5, 4], [-6, 0, -2, -13], [-26, 0, 2, -6], [24, -8, 0, -5], [-22, -10, 6, 4], [5, 5, -3, -5], [41, -9, 3, 8], [-32, 3, 0, -11], [-25, 6, 3, -3], [14, 12, 4, 4], [38, 0, -6, -10], [-18, 4, 4, 6], [-20, 10, 6, 0], [7, -6, 1, 0], [-38, 2, -8, 6], [28, 2, 4, 1], [-7, 9, -3, -4], [-40, -8, -6, -6], [-8, 6, 0, 5], [69, 1, 1, 0], [-4, -9, 0, -1], [14, 4, 6, -6], [7, 0, -5, 1], [22, 1, -4, 3], [-13, -9, 13, 1], [-9, -7, -3, 12], [-4, 16, 4, -4], [-11, 1, -3, -10], [-21, -13, -7, 2], [-7, -17, 1, -5], [9, -5, -9, -14], [23, -9, -1, 10], [12, 16, 12, 4], [3, 7, -1, -6], [-10, 8, -12, -19], [-36, 0, 8, 8], [-10, 21, -14, -5], [17, -3, 1, 6], [0, 5, -12, -3], [-8, -1, -8, -11], [27, -2, 7, -1], [-16, -4, -14, -6], [-11, -3, 3, 4], [3, -18, 15, 3], [44, -10, 2, 6], [18, 4, -10, -2], [1, -2, 9, 12], [-27, 13, -3, -8], [21, 5, -13, -16], [-17, 9, 5, 0], [-12, 0, -4, -8], [-4, -7, -2, -7], [17, 2, -11, -3], [57, 1, 5, 0], [31, -8, -7, -3], [65, 1, 3, 8], [4, 4, 20, 12], [14, 18, -2, 2], [-1, -3, 3, 11], [-29, 5, -1, -5], [22, 6, -4, -4], [-14, 1, -2, 2], [42, 8, -12, -6], [53, 11, -3, 3], [30, 0, -8, -14], [-2, -10, 2, -5], [-6, 5, 8, 5], [-2, -5, 6, 5], [-3, -6, -7, -1], [-1, 13, 13, 0], [8, -10, 0, 8], [36, 1, 14, 3], [-49, -4, 5, 1], [-21, 3, 7, -8], [30, 1, 8, 15], [52, -7, 4, 6], [0, -4, -6, 5], [14, 16, 4, 0], [-5, -5, -3, 0], [17, 3, -1, -11], [-5, -9, -9, 0], [-48, 2, 0, 6], [-8, -19, 2, 2], [8, 5, -6, 3], [-60, 6, -4, -6], [-33, -9, -7, -11], [28, -10, 8, 2], [26, 12, 8, 6], [35, 9, -13, -4], [-32, -17, -16, -8], [30, 2, -12, -8], [-18, -7, 6, -7], [-5, -6, 9, -1], [26, -4, -10, 2], [4, -4, -2, 8], [-27, 9, -5, -15], [-59, 3, -7, 8], [34, 7, -6, -9], [-34, 8, 10, 4], [37, 0, 13, -7], [1, 9, 1, -7], [50, -12, 0, 2], [24, -18, 2, 12], [1, 19, -15, -3], [-68, -12, -4, -8], [-35, 5, 5, -6], [34, 2, 6, -8], [14, 8, -6, -10], [-21, 4, -3, -1], [-5, 7, 19, 4], [-19, -10, 3, -5], [16, -11, 0, 5], [27, -3, -1, 4], [12, 14, -6, 4], [-42, 11, -2, -11], [3, -1, -15, -14], [35, 3, 5, -16], [-3, 8, 5, 1], [39, -3, 1, 5], [13, -7, -19, -11], [-50, -2, 10, 10], [44, -11, 2, 11], [17, 6, -3, 3], [66, -8, 8, 0], [12, -13, -4, 8], [21, 14, 21, 0], [28, 11, -14, -7], [-19, -14, 11, 3], [0, 3, 2, -10], [58, -4, 8, -6], [-4, -14, 4, 14], [4, -8, -8, -8], [3, 7, 7, 12], [21, 5, -7, 6], [-1, -9, -7, -13], [48, 4, -8, -14], [1, -18, 9, -7], [4, -6, 2, -6], [26, -10, 14, 0], [31, -1, 9, 4], [1, -4, -15, -12], [18, -10, 14, 0], [5, 5, -3, -4], [-43, -15, -3, -1], [-11, 5, 3, 2], [14, -2, -18, -6], [7, 11, -21, -12], [12, 4, 10, 10], [45, 1, 11, -9], [-8, 11, 10, -9], [22, 8, 14, 6], [16, -5, -8, 1]];
    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_cb();".
// 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_cb(:prec:=4)
    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_cb();".
// 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_cb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![8, 6, -19, 0, 1]>,<11,R![80, -12, -22, 0, 1]>,<13,R![296, -56, -38, 2, 1]>,<17,R![1024, 58, -71, -2, 1]>,<19,R![256, 112, -26, -6, 1]>],Snew);
    return Vf;
end function;