
// Make newform 5610.2.a.bm in Magma, downloaded from the LMFDB on 07 May 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_bm();".
// 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_bm();".
// 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_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], [-1, 0], [-1, 0], [-1, -1], [-1, 0], [1, -1], [-1, 0], [-1, -1], [1, 1], [0, 2], [-2, 2], [2, 0], [10, 0], [-3, 1], [-4, 0], [-5, 1], [-3, 1], [-2, 0], [-4, 0], [-9, -1], [3, 1], [7, -1], [4, 0], [-4, 2], [0, 2], [0, -2], [8, 0], [-10, -2], [-4, -2], [-11, -1], [8, 0], [-4, 0], [12, -2], [-12, 0], [18, 0], [-8, 4], [-8, 2], [0, 4], [-10, 2], [0, -2], [-5, -1], [-8, 2], [16, 0], [-23, -1], [-10, -4], [10, -2], [8, -4], [-12, 4], [-10, -2], [8, 2], [8, 2], [-12, -4], [25, -1], [15, 3], [8, 2], [24, 0], [2, 0], [10, -2], [-4, 6], [-4, -2], [8, 4], [-2, 4], [-9, -5], [-25, -1], [-24, 2], [-16, -2], [-20, 0], [-9, -3], [-6, 2], [23, 1], [-16, 2], [16, 0], [16, 0], [13, -5], [2, 2], [6, 2], [11, 5], [8, 6], [13, -1], [-12, 2], [-4, 0], [-10, 0], [-22, 2], [-4, 2], [-1, -1], [-33, -1], [7, 1], [18, -4], [-10, 4], [0, 0], [-33, -1], [-4, -4], [20, 4], [-10, -6], [6, -2], [-24, 4], [11, -3], [13, 7], [15, 3], [-6, -4], [2, 6], [14, -4], [16, 4], [-10, 0], [-28, 0], [0, -2], [-17, -1], [8, -6], [-8, -8], [11, -7], [-29, 3], [-35, 3], [3, 5], [18, -6], [-20, -4], [-23, -5], [8, 4], [-8, -4], [-2, -4], [2, 6], [12, 6], [25, -1], [-14, 0], [-4, -8], [-38, 2], [-46, 0], [-34, 0], [37, -3], [8, 8], [9, -1], [39, -1], [-44, 0], [-24, -4], [-10, 0], [-40, 2], [18, -8], [1, -9], [-22, 6], [27, 1], [8, -10], [-6, -2], [-6, -4], [-6, -6], [-14, 2], [34, 4], [3, 3], [32, -6], [10, 4], [8, -4], [30, 2], [-24, 2], [-33, 1], [-42, 2], [-22, 6], [12, 0], [-17, 7], [-8, 12], [39, 1], [14, 0], [18, -4], [-18, -6], [0, 10], [-4, -12], [-15, 5], [12, -2], [-19, -3], [8, 4], [-2, 8], [39, -3], [0, 6], [-30, -2], [4, -10], [-4, 12], [-6, -4], [-4, -4], [33, 3], [-14, 6], [20, 10], [16, 8], [5, 7], [16, -8], [20, 0], [-8, -2], [20, -2], [12, 0], [31, 1], [42, 4], [-4, -8], [-9, -11], [-36, -4], [12, -10], [-22, -6], [6, 10], [35, -3], [14, -10], [-11, 7], [-5, 1], [0, -2], [2, -4], [36, 4], [2, -4], [47, -1], [-4, -2], [-10, 4], [-9, 3], [23, -3], [-12, -4], [6, 2], [28, -6], [14, 2], [-15, 7], [-47, -5], [36, 4], [37, -3], [10, -6], [-50, -4], [6, 6], [-2, -16], [-3, -3], [-23, 7], [23, 1], [-7, 1], [18, 8], [10, -14], [-26, 10], [-1, 9], [21, -9], [7, -9], [6, 6], [-12, 0], [-30, -4], [-8, -12], [-48, 4], [36, 6], [-38, -2], [24, 0], [-10, -4], [8, -2], [-8, -4], [-40, 0], [-16, 12], [11, -13], [24, -8], [0, 2], [4, -10], [34, 10], [18, -6], [-52, 0], [19, 3], [-20, -4], [9, -1], [-2, 4], [-24, -12], [-24, -10], [30, -4], [-6, -2], [-18, 8], [10, 6], [-28, -6], [10, 4], [21, 5], [0, -4], [-54, 4], [-32, -6], [-45, -3], [38, 10], [-1, 9], [22, -12], [-20, 8], [-20, 2], [4, -2], [-24, 4], [30, 12], [-32, 4], [6, 0], [60, 4], [-21, -5], [33, -5], [-12, -14], [8, 4], [0, -8], [-46, -2], [8, -4], [-9, 1], [-2, 2], [18, -6], [60, -2], [-4, 10], [0, 8], [18, -12], [-18, -4], [8, 12], [-14, 12], [-64, -4], [-70, -4], [-36, -10], [-12, -4], [11, -7], [-4, 8], [-53, -1], [-40, 10], [22, -12], [25, 1], [42, 6], [-62, -6], [-26, -8], [-11, 5], [8, 18], [24, 8], [-69, 1], [0, -12], [-18, -4], [10, -8], [-48, 4], [0, 16], [8, -2], [13, 17], [-12, -12], [24, -14], [20, -2], [-16, -12], [30, -8], [-42, -4], [-8, 0], [-48, 10], [-16, -2], [62, 6], [0, 12], [0, -4], [-18, 4], [4, 6], [16, 14], [-29, 11], [-12, 8], [-2, 10], [-38, -10], [40, -6], [-32, 2], [44, 2], [28, -12], [60, 6], [-16, -6], [-30, -12], [-32, 8], [-53, 9], [-28, -16], [-30, 4], [-16, -4], [-52, -4], [-12, -6], [31, -1], [-2, 0], [-9, -7], [-9, 15], [43, -3], [-10, 0], [-26, 6], [-16, -12], [30, 8], [2, -10], [-42, 8], [2, 0], [-6, -10], [-12, 8], [-14, -10], [8, 14], [-10, -4], [-10, -18], [-37, 9], [21, 1], [-38, 2], [-66, 2], [8, 6], [22, 10], [36, -6], [-18, -6], [52, 4], [-12, 6], [54, -8], [-58, 0], [22, -12], [56, -6], [33, -7], [38, -8], [-28, -8], [-47, 9], [-32, 0], [-10, 16], [50, 6], [-26, 2], [-66, 4], [68, -6], [-21, -9], [36, 0], [-28, -12], [13, 11], [36, -8], [14, -12], [-30, 10], [4, 2], [42, -4], [4, 14], [-20, -12], [-67, -1], [-34, -4], [3, 19], [-38, 12], [38, 0], [40, 12], [-74, -6], [-37, -15], [78, 4], [35, -13], [16, 12], [-8, 10], [44, -6], [-16, 16], [-12, -12], [-27, 15], [72, 8], [15, -15], [28, 6], [31, -1], [26, -10], [40, 2], [41, -1], [16, -4], [6, -12], [-74, -6], [82, -6]];
    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_bm();".
// 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_bm(:prec:=2)
    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_bm();".
// 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_bm( : 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, 2, 1]>,<13,R![-16, -2, 1]>,<19,R![-16, 2, 1]>,<23,R![-16, -2, 1]>],Snew);
    return Vf;
end function;