
// Make newform 5610.2.a.br in Magma, downloaded from the LMFDB on 13 January 2026.
// 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_br();".
// 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_br();".
// 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, -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 := 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]];
    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], [3, 1], [1, 0], [-3, -1], [5, -1], [2, 0], [2, -2], [0, -2], [0, 4], [5, 1], [0, 4], [-1, 3], [11, -1], [-6, -4], [-4, 2], [3, -1], [-5, -3], [7, 1], [4, 4], [14, 0], [-4, -4], [-6, 2], [-12, 0], [-4, -4], [10, 0], [-1, -9], [-2, 6], [8, 0], [-2, 0], [0, 8], [-8, -4], [4, -4], [-12, 0], [-2, -2], [-6, -2], [-2, 4], [17, -3], [6, 8], [4, -8], [-15, 3], [12, 2], [10, -2], [6, -6], [-14, -2], [4, 0], [0, -10], [-12, -2], [6, -2], [-9, 7], [13, -3], [8, 2], [28, 0], [0, -2], [2, -10], [10, 4], [-4, -6], [-6, -6], [-6, -8], [15, 7], [-13, -5], [8, 8], [2, 0], [-8, 8], [7, 1], [-4, 8], [3, 7], [-24, 2], [-6, -2], [20, -2], [-9, -7], [0, 0], [4, 8], [27, -3], [12, -10], [7, 9], [6, 0], [-6, -2], [14, 8], [4, -6], [6, -12], [-9, 9], [1, 3], [1, 7], [10, -12], [-4, 4], [22, 2], [5, -13], [30, -4], [16, 6], [0, 2], [36, 0], [4, -8], [23, 9], [7, -15], [3, -9], [-2, 16], [-2, 6], [-6, -12], [4, -12], [-8, -2], [-12, 4], [-34, 4], [29, 7], [8, -14], [20, 8], [5, 5], [-7, 1], [3, 13], [5, 5], [16, -12], [28, 0], [-21, -3], [-4, -4], [26, 2], [2, -12], [36, -2], [20, -2], [25, 3], [24, 6], [8, 4], [-24, 8], [-16, 0], [0, -18], [29, -3], [-24, 4], [-21, 13], [1, -1], [-12, -12], [8, -4], [-22, -14], [0, -6], [4, -6], [9, 17], [0, -16], [31, -9], [10, 2], [-50, 2], [6, -20], [6, 12], [-8, 0], [-2, 12], [-1, 15], [-16, -10], [-22, 8], [-12, 12], [26, -6], [-4, -14], [1, 7], [6, -4], [-18, 6], [-12, -12], [-41, -5], [0, 16], [-11, 7], [36, 2], [-30, 8], [4, 0], [-4, 10], [0, -24], [-21, -5], [16, 18], [13, 11], [-40, -8], [-36, 6], [-19, -15], [-20, 2], [20, -6], [0, -2], [-18, 16], [-32, -2], [12, 0], [-33, 5], [-28, 4], [10, 0], [8, 8], [9, 5], [0, 4], [-2, 6], [44, -2], [-24, -2], [0, 4], [15, 1], [-30, 16], [4, -10], [-15, 9], [-54, 0], [-28, 4], [-40, 8], [10, -10], [-33, 5], [-40, 8], [7, -9], [21, -19], [0, -14], [-58, 0], [-20, 4], [26, 16], [23, 21], [-40, 8], [8, -18], [33, 9], [-5, -21], [24, 16], [34, 10], [6, -22], [46, 10], [1, 15], [-11, -1], [-36, -2], [-45, 9], [44, -6], [26, -12], [-6, 2], [-36, -10], [13, -17], [-11, -3], [15, 23], [1, -13], [40, 2], [-2, 24], [-26, -6], [-13, 3], [-25, -1], [51, -9], [2, 2], [52, 8], [10, 2], [30, 2], [18, -10], [-44, -2], [2, 12], [12, -4], [-18, -16], [-38, 0], [-22, -2], [20, 8], [42, 10], [-51, -5], [-58, 6], [-16, 2], [0, 2], [-36, -14], [-6, -12], [20, 8], [1, -9], [-4, 0], [3, 1], [20, 0], [2, -6], [-20, 2], [6, -24], [36, -6], [0, -14], [-22, -22], [28, -14], [16, -10], [-33, 3], [-32, 12], [-30, 4], [38, 8], [21, 17], [38, -18], [31, 1], [-12, 6], [24, -16], [4, -6], [-14, 24], [4, -10], [30, 16], [-44, -12], [-30, -12], [16, -16], [45, 15], [1, -23], [-18, 16], [18, 16], [4, -16], [14, 14], [-8, -20], [-41, 7], [-12, 12], [-60, 0], [-8, -32], [2, -12], [36, 12], [-32, -10], [-6, 28], [62, -2], [26, -8], [-50, -10], [44, -18], [-2, -12], [-44, -8], [-5, -5], [74, -2], [39, 11], [58, 0], [-54, -4], [-35, 11], [-34, 22], [8, -8], [50, -6], [-37, -7], [6, 8], [-2, 10], [-27, 23], [-44, -12], [20, 26], [-20, 0], [12, -26], [32, 20], [16, -6], [3, 15], [-10, -12], [-28, -20], [-28, -18], [12, 28], [-62, 2], [-48, 12], [-16, -28], [-20, 6], [20, 10], [-52, 4], [-54, -6], [-68, -8], [-16, -14], [-6, 24], [-8, 14], [-1, 13], [-48, 20], [-10, -22], [26, -6], [-40, -18], [-24, -6], [68, 6], [-20, 4], [24, 16], [-12, -18], [50, -16], [68, 8], [-9, 3], [-48, -20], [-8, 10], [12, -2], [18, -12], [-42, 12], [-19, -13], [54, 10], [-41, 17], [29, 25], [-41, 11], [-6, 20], [20, -8], [-66, 10], [18, 12], [-32, 28], [12, 6], [44, -8], [-38, 18], [2, -26], [26, 4], [-34, 12], [0, 2], [-2, -10], [-3, 13], [47, 7], [-28, -20], [26, 10], [-2, 22], [-42, 18], [68, -6], [-44, -24], [-42, 0], [0, 26], [16, 22], [56, -18], [22, -8], [16, -6], [-33, 23], [18, 0], [34, -26], [-15, -1], [28, 12], [58, 2], [-30, -6], [-8, -12], [30, 24], [-14, -16], [-15, 5], [-4, 38], [-42, 10], [29, -21], [-38, -10], [28, -2], [80, -4], [-6, -14], [34, -12], [-30, -20], [10, 18], [-13, -13], [24, 8], [-89, 1], [-10, -18], [-8, 14], [12, 2], [-2, 10], [-29, -15], [14, 12], [81, -5], [-8, 8], [-54, 0], [18, 32], [2, 20], [-8, 6], [-69, -13], [12, -36], [11, -15], [48, -2], [-29, 1], [44, 2], [-4, 20], [-47, -11], [-16, -36], [4, -8], [-22, -18], [0, 18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5610_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5610_2_a_br();".
// 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_br(: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_br();".
// 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_br( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5610_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, -2, 1]>,<13,R![4, -6, 1]>,<19,R![4, 6, 1]>,<23,R![20, -10, 1]>],Snew);
    return Vf;
end function;