
// Make newform 1110.2.a.s in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1110_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_1110_2_a_s();".
// 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_1110_2_a_s();".
// 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, -7, 0, 1], [-2, -11, 0, 1], [-6, 7, 2, -1]];
    Rf_basisdens := [1, 2, 2, 2];
    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_1110_a();"
function MakeCharacter_1110_a()
    N := 1110;
    order := 1;
    char_gens := [371, 667, 631];
    v := [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_1110_a_Hecke(Kf)
    return MakeCharacter_1110_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, 0, 1, 0], [1, -1, 0, 0], [-1, 1, 0, -1], [1, 0, -1, 2], [1, 0, -1, -1], [-1, 1, -2, 1], [0, -1, -1, -1], [0, 1, -1, 1], [-1, 0, 0, 0], [-2, -1, 1, -1], [-2, 1, 1, 1], [0, 1, 1, -2], [-5, 0, 1, -2], [-2, 0, -2, 0], [0, -2, 2, -1], [2, 0, 2, -2], [-6, 2, 0, 2], [-1, -1, 2, 1], [4, 0, 0, 0], [1, -3, 0, 1], [-3, 3, 0, 1], [2, -2, 0, 3], [-6, -1, 3, 0], [0, -1, -1, 2], [3, 1, 2, -1], [1, 3, 2, -2], [-8, 1, 1, 1], [-3, 3, -2, -1], [-4, -2, -2, -2], [-2, -3, -3, -2], [8, -3, 3, -3], [-2, -1, -5, 2], [3, -3, 2, -1], [2, 1, -5, 1], [-1, 0, -5, 4], [9, 1, -4, 7], [-7, -2, 1, 0], [0, 0, 0, -4], [2, 0, 4, 0], [-5, 0, 3, 2], [-6, 1, 1, -2], [-5, -1, 2, -1], [8, 2, -2, 0], [4, 1, -1, 3], [2, -3, 5, -3], [10, 3, 3, -3], [6, 0, -4, 2], [-10, 1, -3, 4], [-6, -3, 5, -1], [2, 4, 4, -4], [-2, -4, -2, 6], [7, -1, -2, -1], [-6, 2, 2, 1], [5, 0, 5, -3], [4, 2, -2, 0], [-19, 3, 0, -1], [-7, -1, 0, 1], [1, 3, -2, -1], [-17, 0, 1, 0], [10, -2, -4, 0], [-2, -1, 3, -5], [-8, -1, 1, 0], [8, -1, -1, -3], [8, 1, 1, 2], [-13, 5, 0, -1], [8, -2, 2, 0], [0, 2, -4, 8], [-4, 3, -5, 7], [-4, -2, -2, -2], [1, 0, 1, -8], [-6, 0, 0, 0], [18, 2, 4, -2], [-11, 1, 0, -1], [20, 3, 3, -3], [-12, 2, 0, -2], [5, 3, 0, 1], [12, 0, 6, -2], [17, 0, 3, -3], [6, -3, -3, 2], [-15, 4, 1, -4], [5, 1, -2, 3], [10, -5, 3, 1], [8, -6, 2, -2], [18, 0, 0, -2], [-18, -2, -4, -1], [-4, 3, 3, 5], [16, -3, 1, -4], [6, 5, 5, -1], [7, 3, 0, 1], [10, -3, -1, 6], [3, 0, 1, -1], [3, -2, -1, -1], [4, -4, 0, 0], [3, 2, -7, 7], [-2, -1, -3, -3], [12, 0, 4, -4], [17, -2, -1, -5], [-21, 6, -3, 4], [4, 2, -4, 6], [12, -5, 9, -5], [11, 5, 4, -5], [12, -7, 7, -5], [-10, -1, 3, 2], [14, 3, 3, -5], [-20, 7, 1, 0], [-8, -2, -10, 4], [17, 4, -1, -4], [6, -1, -3, -6], [4, 5, 1, 1], [6, 5, -3, -6], [18, 1, -3, -1], [2, 1, 5, 3], [2, 7, -3, -5], [21, 4, 1, -2], [-1, -3, -6, 1], [12, 0, -2, 4], [8, -3, 5, -2], [3, -5, -4, 2], [-5, -3, 12, -7], [3, 7, 2, -5], [-10, 1, -11, 5], [0, 1, 3, 3], [16, -8, -2, -6], [-21, -3, 2, 0], [-2, -6, 0, 4], [-18, -1, -3, 6], [8, 5, -11, 5], [8, -1, 5, -3], [-8, 0, 2, -5], [20, 2, 6, -4], [-27, 1, 2, 5], [24, -1, -1, 1], [-2, 4, 0, -8], [21, -4, 7, -2], [10, 0, 2, -6], [2, -6, 2, -10], [-3, 5, -2, 7], [-8, 4, 0, -4], [-9, 0, -1, -3], [29, 1, -4, 3], [-6, -1, -9, 1], [-11, 3, 6, 6], [-2, 0, -10, 0], [-5, -1, 2, 3], [11, -2, -1, 6], [-13, 0, 5, -3], [-26, -2, -2, 7], [-18, 1, 7, 5], [18, 1, -9, -1], [-9, 4, 3, -6], [-12, 4, -6, 3], [1, -1, 2, 7], [0, 2, 10, -4], [-4, 2, 2, 8], [8, 1, 13, -9], [-6, -4, -8, 10], [-18, 9, -11, 6], [10, 1, 1, -7], [-6, -5, 3, 4], [22, -5, -3, 0], [14, 6, 2, 5], [-3, 4, -1, 10], [44, 2, -4, -3], [2, -7, -7, 1], [-31, -1, 4, -3], [-8, -7, -7, -3], [4, -6, 4, -6], [-14, 4, -2, -4], [-4, -6, 10, -5], [-2, -5, -1, 3], [-9, -5, -10, 9], [-9, 5, -2, -5], [22, -6, 10, -8], [0, 1, -3, 8], [-33, -2, 5, 1], [6, -7, 9, -9], [-14, -4, -4, -6], [-16, 1, -7, 2], [14, 4, -2, 6], [15, -7, -4, 13], [-16, -3, 5, -11], [-22, 0, 4, -13], [-15, -4, -15, 7], [-22, 3, 1, -1], [1, 0, 1, -2], [0, -4, -10, 14], [-12, -4, 8, 0], [-2, -5, -9, 2], [-4, 10, 2, 4], [-35, 0, 3, 7], [-11, -6, -1, 9], [-16, 4, 4, -6], [-4, 1, -5, -4], [28, -10, 0, 0], [-22, 4, -8, 10], [8, 3, -7, 0], [-31, 7, 2, -1], [-14, 8, -12, 4], [-17, 3, -12, 3], [-26, -3, -5, 1], [4, -7, -3, 1], [-18, 2, -2, 5], [30, 4, -4, -4], [11, -5, -8, -5], [-9, -1, -4, 11], [9, -3, -6, 3], [22, -11, 1, -3], [12, 2, 2, -3], [14, -5, 13, -3], [20, 5, -11, 4], [3, -7, 2, 3], [-25, 12, -9, 8], [-1, -4, -3, -12], [18, 3, -3, 6], [-18, 0, 4, -14], [23, -7, -6, 11], [1, 10, 9, -2], [14, -4, 8, -2], [11, 3, 8, 3], [16, -11, 5, 5], [10, 7, -11, 4], [36, -4, 4, -4], [-3, 6, 11, -5], [20, 3, -7, -2], [-12, -2, 2, 4], [2, 3, 15, -9], [-8, 2, -2, -2], [32, -5, -1, -9], [4, -6, 10, -4], [-7, 5, -4, 13], [-12, 4, 2, -6], [4, -10, -6, 10], [-28, 2, 0, 11], [41, 1, 6, 1], [-2, 4, -12, -4], [-6, 12, -2, -8], [24, 8, 8, 0], [2, -1, -5, -1], [-11, 0, 7, 1], [37, -9, 2, -1], [-25, 3, 8, -10], [24, 3, 9, -8], [-5, -2, -15, -2], [6, 3, -3, 8], [-6, -6, 16, -10], [32, 1, -5, 3], [-39, -1, 6, 1], [-1, 5, 0, -11], [8, -2, 0, -2], [8, 0, 14, -11], [-20, 6, 4, 6], [22, 4, -4, 12], [2, 2, 10, 1], [18, 8, -12, -4], [34, 0, -10, 12], [31, 0, 5, -2], [-24, -6, 10, -1], [6, 1, -1, -6], [16, -10, 10, -4], [-4, -4, -6, 0], [-16, -5, -9, 7], [31, 1, 8, -1], [-10, 3, -1, -3], [-15, -2, 7, -11], [50, -4, 4, -6], [22, -10, -4, 2], [10, -4, 8, -4], [21, -6, -9, 7], [19, 0, -13, 6], [64, -1, 1, 0], [-2, 1, -7, 3], [17, 9, -10, -7], [-29, 4, -9, 8], [-11, -5, 2, -11], [26, 7, 3, 6], [9, -4, -1, -12], [10, 4, 6, -8], [10, 9, -9, 0], [-16, 11, 5, -13], [8, 2, -4, 15], [28, 6, 4, 4], [12, 2, -2, 12], [42, -2, -8, 16], [36, 0, -8, 9], [-4, -3, 9, -19], [30, -14, 0, 2], [41, 5, -6, 7], [-12, 5, -11, 9], [14, 2, 0, 14], [10, 5, -7, -10], [3, -13, 2, 2], [-2, -3, 3, -11], [6, -1, 3, -18], [3, 3, 0, -9], [17, 5, -6, -3], [8, -14, 2, -8], [-36, 0, 0, 4], [-16, 2, 6, 17], [-26, 0, 4, 8], [-8, -6, 2, -4], [10, -4, -10, 10], [-53, -2, -1, 3], [10, -2, 4, -11], [-50, 11, -1, -3], [-14, 3, 7, -2], [18, 4, 6, 4], [-21, -5, 10, 1], [-8, -9, -1, -6], [43, 2, -7, 7], [0, 3, -5, 5], [44, -8, -4, -6], [-51, 1, -4, 1], [-14, 8, -4, 14], [47, 3, 2, 8], [-1, -3, 2, 17], [2, -12, -12, 20], [-9, -1, -2, 17], [-13, -10, -1, 15], [15, -1, -2, 3], [-26, -7, -9, -1], [-55, 2, -1, 2], [-28, -9, 5, 6], [28, -8, 2, -8], [-2, 7, -13, 9], [-10, 9, 5, 3], [-39, -3, 4, -11], [-4, -11, -3, 15], [20, -6, -12, 14], [-32, 2, 8, -10], [-20, -3, 7, 1], [6, 8, -2, -14], [-21, -10, -5, 9], [12, 2, 14, -2], [14, 6, -2, -8], [6, -9, 15, -18], [-22, 6, -2, -4], [8, -12, 0, 8], [0, -6, -4, -2], [-50, 1, 1, 6], [8, -6, 8, -2], [6, 5, 5, 7], [20, -2, 0, -18], [6, -6, 0, -20], [-28, -8, 14, -8], [-46, 4, -6, 4], [22, 1, 5, -1], [-24, 1, 1, -7], [-40, 3, 5, -5], [5, -3, -14, -9], [-30, -3, 3, -3], [-2, 11, -11, 0], [10, 4, -8, 4], [-4, 7, 9, -8], [19, 0, -17, 12], [-6, 12, -2, -8], [-4, 1, 3, -8], [8, 12, -4, 12], [-18, -6, -10, -4], [-34, -4, 8, 8], [-11, 7, -6, 9], [-40, 6, -6, 2], [46, 6, 0, -8], [-52, 10, -10, 7], [32, -6, 0, 2], [22, 9, -1, -6], [-14, 0, 0, -4], [-20, 4, -8, 12], [16, 5, 17, -4], [-12, -12, 10, 3], [9, 6, -13, 19], [4, 3, -15, -1], [-16, -5, -1, 11], [7, -2, -21, 19], [-14, -6, 8, -2], [-21, -1, -18, 9], [59, 7, 2, -7], [31, -7, -4, -7], [-32, -5, 11, -21], [7, 9, 0, 15], [-6, 3, 1, 8], [-22, -1, 15, -8], [-28, -3, 13, -8], [50, 0, 0, -7], [27, -3, 10, -13], [2, -12, 12, -4], [-30, -3, 13, -9], [-27, 1, 0, 15], [23, 0, -7, -8], [-46, 10, 2, 2], [36, 12, 4, -10], [51, 0, -5, -10], [-8, -6, 10, 0], [-21, 1, 16, 1], [-2, 4, 2, 14], [10, 13, -1, -11], [-28, -1, 13, 3], [-33, 2, -3, -9], [24, -8, 14, -10], [-15, 2, 5, 8], [16, -7, 5, -8], [10, 4, -4, -4], [10, -6, -2, 12], [-9, 1, -2, 17], [-7, -1, -8, 19], [-10, -5, 1, 17], [-24, 0, 12, -12], [2, 0, 4, -5], [-7, -1, -16, -5], [-9, -2, 7, 16], [-28, 3, 13, -3], [-13, 16, -11, -3], [-26, -2, 8, 9], [3, 0, 7, 3], [-4, -8, 16, 2], [31, 4, -13, -4], [30, -9, -5, 12], [-4, -2, 6, -10], [-46, 4, 8, -8], [-25, 15, -22, 9], [-2, -7, 3, 8], [61, -11, 10, -4], [3, -3, 4, -3], [68, -15, 5, -7], [-1, -3, 6, -3], [32, 7, -21, 7], [-30, 10, -4, 2], [26, -5, -13, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1110_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1110_2_a_s();".
// 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_1110_2_a_s(:prec:=4)
    chi := MakeCharacter_1110_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_1110_2_a_s();".
// 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_1110_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1110_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-16, 40, -13, -4, 1]>,<11,R![-40, 68, -23, -2, 1]>,<13,R![328, -44, -38, 3, 1]>],Snew);
    return Vf;
end function;