
// Make newform 5610.2.a.cc in Magma, downloaded from the LMFDB on 25 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_cc();".
// 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_cc();".
// 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, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-1, 2, 0], [-6, -2, 4]];
    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_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], [-1, 0, 0], [1, 0, 0], [-1, -1, 0], [-1, 0, 0], [-1, 1, 0], [-1, 0, 0], [0, 0, -1], [-2, 2, 1], [-6, 0, 0], [3, 1, 1], [-4, -2, 0], [-8, 0, 0], [2, 0, 1], [1, 3, -1], [1, 1, -2], [-5, -1, 0], [0, 2, 0], [0, 2, -2], [-6, 0, 1], [-3, -5, 0], [-4, 0, -3], [2, -2, 2], [0, -2, 2], [-5, 1, -1], [-3, 3, 1], [-2, -2, -2], [4, 0, 2], [-6, -4, 2], [2, 0, -1], [-2, 2, -2], [-4, -4, 0], [0, -6, 0], [-4, -4, 2], [-5, -3, 1], [-7, -1, 1], [4, -4, 2], [5, -1, 3], [3, 1, -3], [-2, 0, -2], [7, -1, -2], [-6, 4, 2], [-5, 5, 3], [-7, 3, -2], [-7, 1, -1], [9, -5, -1], [-4, 4, 0], [5, 3, 3], [-3, -5, 3], [-8, -2, 0], [2, -4, 2], [-1, 5, -3], [-12, 2, -3], [-9, 3, -2], [-4, 2, 2], [7, 5, -1], [2, 4, -2], [-1, -7, -3], [3, 7, -1], [-10, 4, -2], [11, 1, 3], [-8, -2, -2], [4, -2, 1], [0, -2, -1], [-1, -3, -1], [-22, 0, 0], [12, -4, 0], [1, 7, 0], [11, -3, -3], [-11, 1, 2], [-10, 0, 0], [-13, -3, 1], [-4, 6, 4], [-1, -3, 0], [12, -8, -2], [7, 5, 5], [4, 0, 5], [3, 3, -1], [-4, -12, -1], [4, -6, 2], [0, 8, 2], [-16, 6, 2], [-11, -3, -3], [23, 3, 1], [0, -4, -3], [-6, 6, 1], [-16, 0, -7], [-13, -1, 3], [9, -5, 3], [-1, -7, 1], [0, -4, 3], [-3, 9, -1], [12, 2, 4], [3, -1, 1], [-9, -7, 1], [7, 1, -1], [14, -2, 3], [-22, -2, -3], [-4, 2, -7], [12, 6, 4], [-21, 1, -1], [-8, 10, 2], [28, 4, 0], [-21, 7, 3], [10, -2, -4], [23, 3, -3], [-6, -10, -3], [13, 1, -3], [-13, -11, -1], [2, 0, 5], [-19, -7, 2], [31, -3, 0], [10, 4, 1], [9, 7, -5], [13, 7, 1], [-6, -2, -3], [20, -4, -4], [-7, 3, 5], [2, -4, -4], [-7, -7, -1], [34, 0, -2], [39, -3, 0], [3, -1, 1], [12, -8, -4], [-2, 6, -4], [-13, 9, 5], [19, 11, -3], [-12, 6, -3], [20, 0, 4], [13, 7, 2], [8, -4, 3], [3, 1, -1], [-11, 3, -7], [-26, 2, -2], [-16, -2, 0], [14, -8, 2], [11, 7, 2], [-5, -3, 5], [9, -3, -6], [-2, 2, 4], [16, 0, 0], [-9, 3, -3], [4, -10, -6], [6, -10, 4], [-8, 2, -2], [-24, 6, 3], [-5, -1, 5], [-28, 2, -2], [38, 2, -2], [-15, -13, 1], [-37, -1, -3], [-10, -6, -1], [30, 8, -2], [21, 11, -5], [34, 6, -2], [4, -2, -1], [-13, -11, -1], [10, 2, -1], [23, -5, -3], [-7, -7, -5], [4, 0, 0], [-9, 11, -1], [-22, -14, 4], [-1, 7, -4], [0, 6, 6], [-2, -10, 5], [23, 5, 3], [5, -7, 3], [18, -4, 5], [4, -14, -4], [-25, -1, -7], [28, -6, 4], [9, 17, -1], [-17, 11, 5], [19, 9, 3], [-2, 14, 1], [30, 10, 0], [-20, 6, 2], [6, 6, -8], [25, 9, -4], [-8, 4, 4], [5, 7, 7], [5, 1, 1], [-3, 5, -7], [-5, 1, 9], [10, 2, -5], [-15, 5, -7], [38, 0, -4], [-26, -4, -7], [-9, 7, 5], [1, -5, -3], [-6, -14, -4], [30, 2, -4], [0, -8, -3], [26, 2, 8], [2, 16, 7], [-8, 2, -9], [11, 3, -5], [-32, -14, 6], [18, 6, -2], [-7, -15, 7], [-8, 8, -7], [-28, -8, 2], [-2, 16, -2], [11, -9, 2], [13, -7, -6], [20, -4, 0], [26, 2, 0], [0, 12, -2], [12, 4, -2], [11, 13, 0], [-4, -4, 3], [44, -2, 0], [-2, 2, 7], [11, 7, 7], [24, 14, -2], [-4, 12, 4], [27, -5, -1], [-22, -10, 5], [-15, 1, 4], [25, -7, -2], [-36, -4, 3], [-14, -8, -6], [20, 2, 10], [16, -8, 2], [-11, -3, 2], [-24, 2, -3], [-26, 4, -7], [22, -6, 6], [-54, 2, -2], [30, 2, 10], [-24, 16, 4], [-25, -15, 5], [-2, 0, 6], [-28, -18, 4], [18, -10, 2], [2, 12, -4], [-26, 8, 2], [12, -4, -2], [9, 15, -7], [-14, 2, -4], [10, 10, 9], [-25, -11, 1], [-40, 2, 4], [36, 2, 0], [-39, -7, -3], [16, -2, -2], [-40, 4, -4], [-40, -4, 3], [17, -5, -3], [-19, 19, 6], [-14, -2, 2], [23, -11, 3], [28, 2, 8], [-24, 2, -2], [39, 15, -3], [-29, -9, 1], [-15, 11, -7], [-2, -8, -6], [35, -1, 1], [-39, -3, 0], [-36, -16, 4], [-20, 14, 0], [-3, 13, -11], [-2, 4, -11], [18, -6, 0], [16, -4, -3], [-40, 2, 4], [11, 21, 1], [2, -24, -2], [14, -4, -2], [-40, -2, 6], [-10, -16, -8], [23, 9, 7], [-19, -11, 9], [22, -6, -6], [34, -2, -7], [17, -11, 0], [24, 6, 2], [7, 11, -5], [-43, 7, 1], [-11, -5, 3], [5, -21, 1], [33, -7, -6], [-23, 7, 7], [-15, -17, 1], [-33, -11, -5], [6, 20, 6], [12, -20, -4], [-3, -11, -7], [-54, 4, 0], [-2, -18, -8], [-6, -12, 8], [-21, 1, 1], [-9, 7, -3], [-16, -6, 10], [-15, -13, 9], [-3, 1, 10], [37, -5, 11], [36, 10, -7], [7, 15, 11], [4, -10, 2], [-48, 4, -5], [-32, -4, -2], [-8, -12, 10], [11, -11, -11], [-12, -4, -11], [22, -8, 2], [13, -17, -5], [-47, -9, -4], [13, 15, -7], [-20, -6, 4], [24, 16, 0], [-24, -2, -6], [-27, -1, -11], [-8, -14, -4], [3, 15, -8], [-11, -15, 11], [-11, 23, 5], [-32, 14, 0], [-40, 0, -10], [-21, 5, 1], [5, -5, -9], [2, 2, -10], [-8, 6, 0], [-9, -9, 11], [6, -10, 12], [1, -13, 3], [45, -5, 5], [-33, -1, -3], [0, 6, 8], [-42, 0, -2], [26, -2, 7], [10, 10, -10], [-2, -22, -2], [4, 16, 14], [4, 18, 12], [-20, 2, 4], [34, -16, 2], [-26, -10, 0], [66, 2, 4], [56, 14, -2], [17, 9, -1], [3, 1, -13], [-63, 1, 2], [-18, 14, 10], [-11, -11, -9], [0, 22, 6], [-23, 1, 11], [34, 0, 8], [-46, -22, 5], [11, -7, 1], [4, 16, -7], [-21, 15, 4], [-65, -1, -4], [-32, 2, 10], [47, 1, -1], [7, 9, -3], [-38, -16, 8], [-65, 5, -3], [45, -7, -1], [-40, 0, 0], [61, -1, -3], [9, 7, -5], [0, 10, 8], [1, -23, -1], [9, -11, -13], [-18, -6, 2], [4, -2, -13], [9, -3, -10], [-2, 22, -4], [27, -11, 1], [-5, -27, 3], [-31, 7, -1], [-9, 19, -1], [2, -6, -2], [-9, -13, 1], [13, 9, 7], [49, 25, -7], [3, 15, 5], [46, 16, -8], [16, -10, -2], [1, -15, 4], [-32, 6, 2], [46, -6, -2], [-20, 0, -9], [-2, -14, 2], [-14, 6, -10], [-11, 23, 5], [-53, -23, 5], [-18, 8, -8], [32, 10, 2], [-59, -7, 0], [-16, -18, 2], [39, 21, -7], [-31, 15, 6], [3, -3, -3], [-34, -12, 2], [45, 11, -5], [-13, -19, 3], [-20, -10, -4], [6, 20, -6], [-32, -8, 0], [18, -8, 3], [-31, 19, 3], [-56, 4, -7], [-36, -4, 8], [9, -15, 9], [30, 28, -4], [50, -10, 4], [31, 5, 2], [-30, 4, 8], [-54, -10, -3], [26, 14, -8], [-41, -17, 7], [16, 18, 2], [-69, -9, 1], [-8, -26, 4], [-4, 6, -7], [-8, -24, -8], [6, 10, -9], [-1, -21, -7], [-36, 12, 5], [61, 1, -5], [65, 3, 5], [41, -11, -4], [-34, -10, -10], [-30, -2, -14], [-32, -8, -14], [-47, 13, 5]];
    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_cc();".
// 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_cc(:prec:=3)
    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_cc();".
// 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_cc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5610_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-8, -8, 2, 1]>,<13,R![-8, -4, 4, 1]>,<19,R![-56, -28, 0, 1]>,<23,R![-344, -44, 8, 1]>],Snew);
    return Vf;
end function;