
// Make newform 5610.2.a.cg in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5610_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_5610_2_a_cg();".
// 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_5610_2_a_cg();".
// 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, 11, 2, -1], [2, -7, -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_5610_a();"
function MakeCharacter_5610_a()
    N := 5610;
    order := 1;
    char_gens := [1871, 3367, 1531, 3301];
    v := [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_5610_a_Hecke(Kf)
    return MakeCharacter_5610_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 := [[1, 0, 0, 0], [-1, 0, 0, 0], [-1, 0, 0, 0], [1, 1, 0, 0], [1, 0, 0, 0], [0, 1, 1, 0], [1, 0, 0, 0], [1, 0, 0, 1], [4, 0, 1, -1], [-1, 2, -1, 0], [-1, -1, 0, 0], [2, 1, -1, 0], [-1, 1, 1, -1], [-3, -2, -1, 0], [2, 0, 0, 2], [1, -1, -1, 1], [0, 2, 0, 2], [-6, -1, 1, 0], [-1, 0, 0, -1], [6, 2, 0, 0], [-2, -2, -2, 0], [7, -1, 1, -1], [0, -3, -1, 0], [4, 0, 2, 0], [0, -2, -1, 1], [3, -1, 1, 1], [7, -1, -2, -2], [-1, 0, -1, 0], [-2, -1, -1, -2], [7, -1, 1, 1], [0, -2, 0, 2], [0, 3, 1, 0], [15, 1, 0, 0], [3, 2, -1, 2], [9, 0, 2, -3], [0, -1, 1, 0], [9, -1, 1, -1], [-3, 2, 3, 0], [-2, -4, -2, 0], [4, -1, 1, 2], [-2, 1, 1, 0], [-3, -2, -1, 2], [3, -4, -3, -2], [6, 2, 2, 0], [10, -1, -1, 2], [-4, 1, -1, 0], [-5, 0, -1, 4], [9, -1, -2, 0], [-7, 2, -3, 2], [-4, -7, -3, 0], [15, 0, -1, 2], [4, 0, -2, -2], [-10, 4, 1, -1], [-10, 3, 1, -2], [11, 0, -1, -2], [5, 3, -2, 0], [0, -1, 1, -2], [-9, 2, 1, 0], [-8, 0, -4, 2], [9, -2, -3, 0], [4, 6, 4, -2], [19, 0, 5, -4], [2, -2, 2, -2], [-2, 0, 0, 2], [14, 0, -1, -3], [-7, -4, 1, -2], [-1, 0, -3, 2], [-22, -2, 2, 0], [10, 0, 0, -2], [-3, 5, -1, -1], [7, -1, 2, 0], [-2, -2, -4, 4], [-1, 3, -1, -3], [10, 0, 0, 4], [-2, -1, -3, -2], [-10, 4, 0, 2], [6, -2, -2, -4], [12, 2, -2, 2], [9, 1, -2, 4], [-2, 2, 0, 2], [-7, 0, 3, -2], [-2, -5, -1, -2], [4, -5, -2, -1], [6, 2, 4, -2], [-11, -1, 3, 3], [6, 1, -2, 3], [-3, -3, -4, -2], [-8, -3, -1, 2], [15, 3, 5, -3], [16, -3, -1, 4], [9, 7, 1, 1], [4, 2, 1, 5], [-7, 1, 5, 3], [-19, -3, -1, 1], [-8, 0, 4, 0], [0, 4, 4, -4], [14, -4, 0, -4], [-10, -1, -3, 0], [-7, -2, -1, 4], [2, -2, 0, 2], [-16, 3, -1, 2], [-5, 2, 5, 2], [8, -5, -3, -4], [18, -1, 1, 0], [6, 0, 0, -6], [-8, -2, -2, 2], [-12, -3, -4, -1], [-6, 4, 0, 4], [-5, 1, 2, 4], [9, 2, 2, -1], [-4, -1, 1, -8], [2, 4, 4, -4], [3, 3, 1, 1], [-12, 5, 1, 2], [-8, -8, -2, 2], [1, 4, 5, -6], [-15, -8, 3, -2], [-10, -2, -4, 4], [-9, 0, 5, 4], [-20, -5, 0, -3], [-6, -7, -7, 2], [-16, 2, 4, 0], [-14, 0, 0, 0], [2, -5, 5, -2], [-12, -5, 1, -4], [-13, -5, -7, 5], [12, -2, 2, 2], [-4, -8, 2, -2], [-11, 2, -1, 0], [6, -3, 1, 2], [-11, 4, 2, -1], [2, 2, -2, -2], [-5, -8, 5, -2], [6, 5, 2, 1], [-13, -7, -4, 0], [-2, 6, 2, 4], [-7, -4, -2, -3], [-16, 7, 1, -4], [-7, 2, 0, 5], [-5, 3, -7, 1], [-4, -2, -2, -4], [-13, 0, -1, 6], [9, -9, 3, -1], [-13, -2, 3, 2], [10, 5, -1, 0], [-16, -2, 4, 2], [3, -10, 1, -2], [1, 5, 4, 2], [-15, -8, 3, -2], [4, 0, 0, 4], [3, -2, 1, -2], [-11, 2, 1, -4], [-3, -5, 1, 1], [-18, 2, -2, 2], [-15, -4, -1, -2], [-8, -8, 0, -4], [13, -3, 4, 0], [11, 3, -2, -4], [-8, -3, -1, 2], [4, -4, -4, -6], [-2, 4, -2, 0], [10, 6, -6, 0], [-10, -2, -4, 4], [-26, 5, 3, 2], [-2, -3, 3, 4], [20, -5, 2, 3], [7, -5, 2, 2], [-27, 8, 1, -2], [0, -1, -4, 3], [-3, -2, -9, 2], [24, -3, 8, -9], [-12, 5, -7, 0], [-27, -1, -7, 5], [17, 1, 0, -6], [10, 2, -2, 2], [-16, 1, -1, 6], [-2, 10, 6, 2], [2, 8, 4, 4], [0, 4, -4, 0], [-6, -7, 0, 3], [11, -1, -2, -6], [14, -5, -3, 2], [38, 5, -1, 0], [6, -8, -8, -4], [-10, 10, -2, -2], [-22, -10, 2, -4], [-33, 0, -1, 2], [11, -3, 3, -5], [8, 1, 4, -7], [-11, 6, 2, -3], [-10, -5, 2, -1], [-24, -10, 2, -4], [3, 4, 3, -4], [15, -4, 3, -2], [-19, 6, -3, 2], [17, -5, -1, 1], [37, -2, -2, -5], [2, 9, -1, 0], [5, -1, -4, 4], [17, -3, -6, -2], [-4, -12, -4, -6], [3, -1, -1, -3], [-14, -7, 2, -7], [-20, -2, -4, 8], [-4, -3, -3, -2], [39, 9, 1, 3], [-20, -6, -4, -2], [8, -1, -3, 0], [19, 1, -3, -1], [29, 0, 1, 4], [-10, 14, 8, -2], [-9, 2, -5, 0], [13, -11, -5, 5], [8, 9, 2, 1], [-15, 6, 2, -7], [22, 2, 8, -6], [-14, 6, 8, 4], [19, 14, 3, 4], [13, -5, -7, -7], [-23, 9, 3, 3], [-8, -5, -2, 1], [-5, 7, 5, -1], [-4, 2, 0, 0], [-41, -5, -5, 1], [-20, 4, -4, 8], [-18, -13, 0, -3], [23, -7, -5, 5], [6, 0, 2, 0], [-26, -2, 0, -8], [4, -5, 3, -6], [-2, -13, -4, 1], [-1, -8, -9, 0], [9, 5, -2, 10], [13, -7, -2, 0], [-5, 9, 7, -5], [37, -4, 3, -6], [2, 10, -4, 2], [-28, -4, -2, 0], [-29, 6, -1, -2], [-36, 1, -5, 0], [6, -6, 10, -2], [-11, 6, -4, 3], [3, 5, -4, -2], [-28, 11, -3, -2], [-7, 4, 9, -2], [22, 3, 4, 3], [-9, 9, -7, 5], [-7, -10, 3, -8], [19, -9, -5, 1], [25, 6, 7, 0], [16, -5, -1, 0], [-17, 5, -3, -1], [4, -4, 0, -8], [26, 10, 6, 0], [31, 6, 5, -6], [8, 5, 8, -1], [52, -2, 2, 2], [16, -1, 3, 2], [41, 4, -3, 2], [-26, -16, 0, -4], [6, 11, -7, 6], [18, -16, -6, 4], [-18, 1, 3, -8], [-19, 2, -5, 2], [3, -13, 3, -1], [-5, -8, 5, -2], [1, -12, -5, 0], [32, -10, 0, 8], [-4, -3, -7, -6], [-11, 10, 1, 0], [-13, 8, -5, 10], [-9, 16, -2, 1], [-8, 6, -2, 2], [7, 1, -2, 4], [-15, -11, 4, -6], [4, -1, 1, -4], [11, -11, -5, 3], [-10, 9, 2, 5], [-12, -4, -8, -2], [-45, -15, -1, -5], [27, 0, 5, -6], [30, -7, -9, 2], [-4, -14, -6, -4], [-18, -1, 4, -3], [14, 5, 5, -8], [-28, 4, 4, -4], [11, 0, 0, 5], [20, -1, 9, -6], [-30, 8, -6, 0], [53, 1, 2, 4], [20, -11, -1, -2], [-6, -18, -6, -2], [-8, -14, 8, -4], [-28, -11, 1, -6], [35, 2, 1, 2], [14, 8, 0, 4], [-22, 8, -2, 4], [5, -4, -6, 5], [-26, -9, -9, 4], [-29, 2, 1, 0], [5, 7, -2, 10], [-15, -14, 3, -6], [-21, -1, 11, 1], [-24, 1, 7, -4], [12, 1, 7, -8], [31, -4, -4, 1], [16, -5, 4, -3], [12, 0, 4, -2], [33, 2, -5, 0], [-32, 1, -1, 2], [-28, 14, 4, -2], [-12, -5, -7, 6], [55, -1, -1, 3], [-15, -2, -6, 1], [-3, 14, 7, 4], [10, -8, 10, -2], [26, 6, -2, 2], [-7, 0, 0, 9], [45, 9, 3, 3], [-25, 3, -6, 4], [-21, -8, -3, 6], [-34, -4, 3, -3], [13, 2, 0, -7], [-11, 13, 10, 2], [-30, -21, -7, -4], [-3, 3, 10, 2], [-32, 0, -12, 8], [25, -2, -1, -12], [-16, -2, 2, 4], [6, 9, 1, 2], [16, -1, 5, 2], [-7, -6, -5, 6], [31, 1, 1, -3], [8, 8, -4, 8], [12, -18, -6, 0], [-12, 7, -3, -4], [-36, -3, -3, -4], [23, -3, -8, 4], [-10, -4, 2, -2], [6, 0, 6, 0], [19, -3, 3, 1], [22, -1, 1, 4], [-21, -14, -11, 6], [54, -2, 4, -4], [23, 0, -2, -1], [34, -8, 2, 4], [-16, -10, -10, 2], [15, -3, -13, -1], [-28, 13, 10, 1], [17, -10, 3, 2], [-13, 12, -8, -1], [-27, 12, 8, -5], [-31, -18, 1, -8], [-16, 8, 8, 4], [-15, 1, -5, -5], [6, -6, 2, -16], [21, 2, 11, -4], [10, -18, -2, 2], [1, 5, -6, 4], [44, 6, 12, -6], [-5, -10, 5, -6], [5, 1, -5, 11], [60, 2, -2, 4], [18, 11, -1, 4], [-27, 0, -8, 5], [-14, -18, -4, -2], [36, -7, -9, 2], [-14, -12, -6, 4], [11, -5, -3, 5], [-24, -15, -3, -2], [-14, 6, -2, -6], [-24, 4, -2, 2], [21, 14, 2, 3], [-1, 1, 4, 6], [-2, -7, -3, -10], [-14, -6, 6, -2], [-10, -10, 5, -9], [49, 5, 0, -2], [-22, -9, -7, 0], [-16, -7, 3, -6], [-10, 3, -1, -6], [-7, 19, 0, 0], [-18, -18, -2, -2], [5, 6, -11, -4], [23, 2, -3, -8], [-20, -5, -10, 7], [-4, 0, -10, 6], [-30, -15, -6, -7], [-6, -4, 10, -8], [46, -8, 0, -2], [14, 8, -10, -2], [-33, -22, -3, -6], [-34, 7, -13, 8], [-21, 0, -6, 9], [62, -5, 1, 2], [-6, -12, -6, 2], [20, -16, -12, 0], [-46, 6, 6, -4], [2, 5, -7, 8], [11, 24, 10, -1], [6, 2, 0, 2], [72, -6, 2, 2], [-34, 2, -10, 6], [1, -7, -3, -3], [-7, -18, -6, -9], [-7, 3, 3, -5], [-9, 11, -7, 1], [-32, -2, -6, -2], [29, -8, 2, 3], [-25, -10, 1, 0], [4, -10, 2, -6], [18, 10, 6, -4], [15, -13, -11, -1], [-49, -2, 3, 6], [-7, 5, 0, 14], [4, 19, 1, 6], [-39, -13, 1, -3], [7, 7, 13, -5], [-11, 1, 5, -7], [-60, 7, 5, 0], [56, -2, 6, -2], [-24, -8, 0, -10], [-3, -2, 2, -3], [31, -19, 3, 7], [23, 2, 2, 5], [49, 8, 4, -5], [-12, -4, -8, -4], [-35, -11, -9, 3], [-47, 10, 3, 4], [46, -11, -4, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5610_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5610_2_a_cg();".
// 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_5610_2_a_cg(:prec:=4)
    chi := MakeCharacter_5610_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_5610_2_a_cg();".
// 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_5610_2_a_cg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5610_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-16, 40, -13, -4, 1]>,<13,R![304, -60, -44, 1, 1]>,<19,R![256, 112, -32, -5, 1]>,<23,R![-1712, 496, 13, -14, 1]>],Snew);
    return Vf;
end function;