
// Make newform 5610.2.a.by in Magma, downloaded from the LMFDB on 20 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_by();".
// 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_by();".
// 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, -3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 2, 0], [-8, -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], [0, 1, 0], [1, 0, 0], [2, 0, 1], [1, 0, 0], [2, 0, 1], [-2, 1, 0], [2, 2, 0], [0, 1, -1], [2, 0, 0], [-4, -1, 1], [4, -1, 0], [0, 0, 0], [4, -2, -1], [-4, -1, 0], [-2, -1, -1], [6, -2, 0], [4, 1, 0], [2, 3, -2], [2, 0, 1], [0, 0, 0], [-2, -2, -2], [4, -4, 0], [-8, 1, 1], [-4, 0, 0], [4, -5, -1], [2, 3, -1], [4, 1, 0], [8, 3, -1], [-8, 4, 0], [-6, -3, 1], [-4, -3, 1], [4, 1, -1], [8, -3, 1], [0, -2, -2], [12, -5, -1], [8, -1, 1], [-10, -1, 1], [-4, -1, 2], [2, 3, 1], [4, -5, -1], [-6, 1, -2], [2, 2, 0], [0, -5, 1], [8, 3, -3], [8, -3, 1], [4, -2, 0], [10, 2, 0], [6, 0, 2], [-12, 3, -1], [8, 0, -1], [8, -1, -2], [10, 1, -1], [0, -3, 3], [-10, 2, 0], [-4, -3, -1], [10, -4, -2], [10, -2, 0], [-8, -4, 0], [-14, -2, -2], [0, 1, 4], [0, -3, 4], [4, 2, -2], [6, 6, 2], [8, 0, 0], [10, -1, -2], [12, 6, 0], [8, -3, 2], [2, 5, 3], [0, 3, -1], [14, -3, -1], [-10, 0, 1], [4, -7, 1], [8, -4, 0], [-2, -2, 1], [18, 1, 3], [26, -1, 0], [-18, -2, -2], [-4, -4, 2], [-10, 0, 0], [-26, 4, 0], [6, 1, 3], [-6, -2, -1], [-10, 5, -2], [6, -3, 0], [14, 0, 2], [-8, 1, -1], [8, 3, 3], [26, 5, -2], [6, -6, -4], [2, 3, 1], [6, 3, -1], [12, -2, 0], [-24, 3, -1], [-10, -4, -1], [-14, 1, -6], [20, -3, 0], [-10, -3, -7], [8, -6, -2], [6, 0, 0], [-16, 2, 2], [6, -3, 1], [20, 3, -1], [6, 7, 1], [14, -7, -2], [2, 5, -3], [-4, 5, 1], [8, 0, 5], [-16, -3, 0], [14, 6, -5], [-4, 5, -6], [-20, 2, -4], [24, 1, 3], [10, -5, -4], [8, 0, 0], [4, 10, 0], [10, 10, -2], [-18, -3, 1], [-10, 6, 0], [-22, -3, 2], [10, -2, 4], [-12, -12, 4], [-4, 3, -5], [-16, -7, -1], [10, 2, 8], [24, 3, 4], [-20, -4, 4], [14, -6, -5], [14, 6, -5], [4, 1, 1], [-16, 9, 1], [20, 6, -4], [14, -4, 6], [-22, -6, 4], [24, 0, -1], [8, -6, -4], [24, 6, 3], [0, 3, -5], [-8, 5, -5], [14, -1, 7], [10, -1, 7], [-12, -3, 1], [-22, -6, -2], [0, -3, 0], [-10, 12, 0], [18, 6, 6], [8, -6, -6], [-12, 12, -2], [-18, -2, 2], [-10, -1, -10], [-10, 6, -6], [-4, 5, -1], [8, 0, -4], [-4, -1, 2], [-32, -1, -1], [10, -11, -4], [2, -6, -6], [6, -3, 5], [-12, 0, -4], [-18, -3, 1], [8, -3, 7], [0, 9, -2], [2, -7, -1], [18, -3, 0], [28, 5, 1], [2, -6, 2], [-28, 6, -1], [-34, -10, 4], [-6, 3, -5], [-14, -4, 2], [-14, -8, -6], [10, -16, -4], [-24, 1, -5], [-10, 11, 0], [20, 3, 3], [6, -6, 6], [-8, 5, 3], [-16, -3, 0], [4, 14, -6], [12, 3, -5], [10, -1, 1], [-18, 15, 3], [0, -6, -6], [-10, 2, 5], [-42, 4, 2], [-14, -4, -2], [8, -6, -1], [-10, -4, 2], [12, -8, 0], [4, 13, 1], [20, 12, -2], [26, 8, 3], [-12, 9, 5], [-12, -3, 0], [-16, 0, 1], [-22, -3, 7], [22, -2, 2], [16, -2, 6], [-10, -13, -1], [14, -6, 3], [20, 12, -8], [-22, 6, -4], [-24, -3, 4], [-32, -2, -5], [-24, -8, 0], [-16, -4, -2], [4, -5, 3], [-20, 2, 0], [14, 9, -2], [-2, -14, 7], [14, -3, -1], [10, -11, 0], [-10, 14, 2], [-10, 6, 6], [0, 1, 1], [-30, 3, 3], [-2, -11, 4], [-8, 10, 1], [-12, 1, -2], [26, 6, 5], [-34, -4, -2], [-10, -15, 1], [-4, 8, -2], [-28, -9, 4], [4, 7, 6], [-28, -1, 2], [-24, -5, 3], [-12, -6, -10], [-36, -2, 4], [-16, 3, -3], [-12, 1, -5], [14, -4, 6], [26, 0, 4], [16, -14, -6], [-34, 0, 4], [14, -1, -3], [20, -4, 6], [20, -7, -3], [-8, -14, 0], [22, -4, 5], [-4, -15, 1], [-10, 6, 0], [-2, -2, -4], [-26, -8, 4], [-26, -7, -3], [36, 0, 4], [-18, 10, -1], [24, 3, -3], [-22, -6, -5], [-28, -7, -5], [-12, 0, -10], [-54, 4, 2], [-2, -2, -6], [34, 7, 7], [-42, 4, -2], [8, 0, 4], [-10, -10, -4], [26, -12, 4], [-12, 13, 2], [-12, -12, -4], [-10, 9, 1], [34, -10, -4], [24, -9, -4], [32, 12, -2], [14, -10, -7], [-34, 2, -8], [-40, 6, 0], [-30, 6, 4], [-10, -3, -9], [26, 6, 4], [14, -18, 2], [-16, -9, -9], [6, 4, -2], [8, -6, -6], [-10, -1, 4], [-36, 1, -4], [-2, -16, 8], [-14, 7, 1], [-8, -11, 7], [-4, -9, -3], [0, 3, 7], [20, 3, 4], [16, -4, 10], [4, 7, -1], [-12, -2, 6], [34, -11, 3], [8, 6, -10], [-26, 13, 5], [-22, 8, 4], [-24, 24, 6], [26, -10, -2], [-12, 15, -1], [-22, 0, 4], [26, 8, 2], [8, 9, -1], [-8, 10, -1], [48, -3, 5], [44, 3, -2], [26, 6, 0], [22, 10, -2], [-10, 0, -5], [-24, 24, 2], [44, -9, -1], [-4, 0, 2], [42, 3, 6], [-14, 5, -1], [20, -1, 5], [-10, 0, 3], [-36, 21, 5], [30, 0, -6], [4, 1, -9], [18, 10, -4], [0, 3, -3], [-18, 4, -2], [44, -1, 8], [-2, -2, 0], [12, 10, -6], [-10, -10, -4], [-28, 15, -1], [-52, 6, 0], [-4, -1, -11], [68, 0, 4], [-26, -8, 10], [-10, 15, -5], [-4, -9, 7], [12, 19, -5], [-24, -3, -15], [38, -10, 0], [2, 9, 1], [6, -6, -12], [30, -14, 3], [12, 6, -10], [0, -5, -3], [8, -4, 10], [-14, -4, 6], [-26, 10, 4], [22, -4, -2], [32, -9, -3], [12, 4, -8], [18, 3, 13], [-2, 19, 11], [40, -13, 1], [-20, 4, 1], [44, -4, 0], [-2, -16, -6], [18, -14, -8], [34, -8, 2], [6, -12, 4], [-10, -18, -1], [-16, 0, 10], [-2, 22, 1], [-12, 13, -8], [-40, -3, 6], [-18, 6, -10], [4, 13, 1], [16, 13, -7], [-42, -6, -6], [16, 10, -10], [-38, -16, 4], [28, -11, 7], [-4, -16, 6], [0, 15, -5], [2, 0, 8], [-10, 21, 7], [-34, 2, 0], [-64, 15, 3], [56, 0, 5], [72, -3, 2], [4, -7, -7], [52, -11, -5], [32, 17, -11], [-4, 21, 3], [54, 13, -5], [-4, -4, 4], [42, 6, 0], [6, 16, 8], [14, 23, -5], [34, 14, -2], [34, -8, 0], [66, -3, -1], [32, -3, -6], [-6, 24, 4], [-40, -3, 7], [18, 15, 6], [-8, -16, -8], [-4, -6, -4], [32, -12, -8], [-32, 4, 0], [-2, -22, 2], [-2, -14, -10], [60, -9, -2], [34, 14, -4], [20, -13, -7], [-34, -15, 6], [12, -5, -3], [-10, -4, -6], [4, -10, -12], [-8, 19, 3], [-34, -9, 7], [10, 7, 3], [-8, 11, -1], [52, 1, 12], [-8, 7, -3], [2, 24, 1], [-4, 18, -4], [30, 15, 7], [-18, 4, -10], [44, 2, -4], [-50, -1, -10], [2, 8, 4], [62, -1, 8], [44, -9, -1], [-14, -4, -6], [-50, 4, 0], [34, -14, 4], [2, 3, -3], [44, -7, 8], [16, 4, 0], [-6, 6, -7], [70, -4, 0], [2, 11, 12], [34, -11, 5], [-28, -6, 2], [-4, -22, 1], [12, -6, 2], [-20, 7, -3], [12, 10, -12], [10, 10, -2]];
    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_by();".
// 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_by(: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_by();".
// 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_by( : 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, -12, 0, 1]>,<13,R![136, -24, -6, 1]>,<19,R![136, -24, -6, 1]>,<23,R![-24, 0, 6, 1]>],Snew);
    return Vf;
end function;