
// Make newform 1110.2.i.o in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1110_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1110_i_Hecke();"
// 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_i_o();".
// 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_i_o();".
// 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, -22, 121, -4, 11, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [2, -11, 0, -1, 0, 0], [86, -2, 11, 1, 1, 0], [0, 121, -2, 11, 0, 1], [0, -471, 8, -44, -1, -4]];
    Rf_basisdens := [1, 1, 11, 22, 22, 22];
    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_i();"
function MakeCharacter_1110_i()
    N := 1110;
    order := 3;
    char_gens := [371, 667, 631];
    v := [3, 3, 2];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1110_i_Hecke();"
function MakeCharacter_1110_i_Hecke(Kf)
    N := 1110;
    order := 3;
    char_gens := [371, 667, 631];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := [[0, 0, 0, 0, -1, 0], [1, 0, 0, 0, -1, 0], [1, 0, 0, 0, -1, 0], [0, 0, -1, -1, 0, -1], [0, -1, -1, 0, 0, 0], [-1, 0, 0, 0, 1, 0], [0, 1, 0, 0, -3, -1], [1, 0, 1, 0, -1, 0], [0, 0, 0, 2, 0, 0], [2, -2, -2, 1, 0, 0], [4, -2, -2, 0, 0, 0], [-2, 1, 1, 0, -1, -2], [1, 0, 2, -1, -1, -1], [0, 2, 2, 2, 0, 0], [0, -1, -1, 2, 0, 0], [0, 2, 0, 0, -1, -1], [0, 3, 0, 0, -2, -2], [-1, 0, 0, -1, 1, -1], [-3, 0, 3, 1, 3, 1], [2, 0, -1, -3, -2, -3], [-2, -2, -2, 0, 0, 0], [-10, 0, 2, 4, 10, 4], [0, -1, 0, 0, -8, -1], [0, -2, 0, 0, -6, 2], [2, 0, 0, 0, 0, 0], [5, -1, -1, -3, 0, 0], [0, 3, 3, 1, 0, 0], [-1, 0, -6, -1, 1, -1], [0, 4, 0, 0, -1, 3], [0, 0, 0, 0, 6, -3], [0, 1, 0, 0, 2, -3], [0, 1, 0, 0, 4, -2], [12, 2, 2, 1, 0, 0], [0, -4, 0, 0, 6, 6], [-8, 0, 0, -1, 0, 0], [6, 0, 2, 4, -6, 4], [0, -6, 0, 0, 1, 0], [0, 3, 0, 0, -9, -1], [6, 0, 5, 2, -6, 2], [0, -4, 0, 0, -3, -1], [6, -2, -2, -2, 0, 0], [6, 0, 0, -4, -6, -4], [9, -2, -2, 1, 0, 0], [3, -4, -4, -7, 0, 0], [0, -8, 0, 0, -4, 2], [-11, 1, 1, -1, 0, 0], [11, -3, -3, -4, 0, 0], [0, -4, -4, 0, 0, 0], [-4, 0, 1, -1, 4, -1], [-5, 0, 0, -1, 5, -1], [15, 1, 1, -2, 0, 0], [0, -1, 0, 0, -3, 0], [16, 0, -1, -4, -16, -4], [2, 3, 3, -2, 0, 0], [0, -4, 0, 0, 18, -3], [2, 0, -8, 0, -2, 0], [13, 9, 9, 4, 0, 0], [0, -1, 0, 0, -13, -1], [-5, 0, 0, -6, 5, -6], [0, -2, 0, 0, -6, -2], [-9, 0, 3, -1, 9, -1], [4, 0, -4, -6, -4, -6], [8, -6, -6, -4, 0, 0], [0, -4, 0, 0, -7, -1], [0, -6, 0, 0, 2, 2], [0, 2, 0, 0, 1, 5], [0, 5, 0, 0, 14, -5], [7, 0, -4, -11, -7, -11], [-17, 5, 5, 2, 0, 0], [0, 0, 0, 0, 12, 4], [0, -1, 0, 0, -7, 2], [7, -1, -1, 4, 0, 0], [3, 0, 1, 4, -3, 4], [5, 0, 9, 2, -5, 2], [0, -1, 0, 0, -6, 3], [2, 0, 3, 6, -2, 6], [1, 0, -3, 8, -1, 8], [10, -1, -1, -2, 0, 0], [-3, -6, -6, -5, 0, 0], [0, 0, 0, 0, -1, 2], [0, -2, 0, 0, -10, -2], [-23, 0, 0, 1, 0, 0], [-8, 0, 5, 5, 8, 5], [16, -4, -4, -4, 0, 0], [-1, 0, 5, 5, 1, 5], [-12, 7, 7, 9, 0, 0], [-4, 0, 0, 4, 4, 4], [16, 0, -2, 2, -16, 2], [0, 0, 0, 0, 8, 11], [15, 0, -3, 2, -15, 2], [-11, 1, 1, 0, 0, 0], [0, 6, 0, 0, 0, -4], [-15, 5, 5, 8, 0, 0], [12, 0, 0, -2, 0, 0], [2, 0, -1, 3, -2, 3], [0, 1, 0, 0, -4, 4], [0, 3, 0, 0, -3, 0], [22, 0, 0, 2, -22, 2], [-14, 0, -8, -2, 14, -2], [9, 0, 0, -3, 0, 0], [-8, -8, -8, 4, 0, 0], [0, 0, 0, 0, 10, 4], [0, 5, 5, 9, 0, 0], [-11, -6, -6, 3, 0, 0], [0, -5, 0, 0, 1, 6], [0, -8, 0, 0, -1, -3], [-8, 0, 11, 9, 8, 9], [-23, 1, 1, 5, 0, 0], [-10, 0, -5, -5, 10, -5], [0, -3, 0, 0, -4, 4], [0, 1, 0, 0, -13, -6], [0, 0, 0, 0, -3, 2], [0, -4, 0, 0, 22, 3], [-7, 8, 8, 9, 0, 0], [0, -3, 0, 0, 1, -9], [0, 6, 0, 0, -19, 3], [-15, 3, 3, -5, 0, 0], [-24, 0, -1, 4, 24, 4], [-12, 0, 6, 14, 12, 14], [22, 0, -3, -2, -22, -2], [-8, 0, 2, -6, 8, -6], [11, 0, -4, -3, -11, -3], [3, 10, 10, -1, 0, 0], [0, -10, 0, 0, -6, 2], [0, 5, 0, 0, -3, 4], [0, 1, 0, 0, 19, 5], [-19, -10, -10, -5, 0, 0], [0, 1, 0, 0, -6, -13], [-31, 0, -1, 4, 31, 4], [2, 0, -7, -6, -2, -6], [-15, 9, 9, -8, 0, 0], [0, 0, -5, 10, 0, 10], [9, 11, 11, -1, 0, 0], [0, 7, 0, 0, -27, -10], [0, 0, 0, 0, 5, -3], [9, -4, -4, -14, 0, 0], [-27, 0, -8, -3, 27, -3], [3, -1, -1, -9, 0, 0], [0, -10, 0, 0, 10, 12], [34, 0, 2, -2, -34, -2], [7, 0, 6, 3, -7, 3], [-4, 0, 14, 9, 4, 9], [0, -11, 0, 0, -12, 7], [5, 0, -9, -10, -5, -10], [0, -6, 0, 0, -15, 7], [0, -4, 0, 0, 39, 5], [0, -5, 0, 0, 18, 6], [-10, -2, -2, -7, 0, 0], [12, -9, -9, -5, 0, 0], [0, 9, 0, 0, -20, -2], [23, -2, -2, -4, 0, 0], [-14, 0, 2, -8, 14, -8], [-15, 0, -5, -5, 15, -5], [22, -2, -2, 2, 0, 0], [24, 0, 4, 8, -24, 8], [-9, 11, 11, 4, 0, 0], [1, -5, -5, -5, 0, 0], [-4, 0, 2, -2, 4, -2], [0, -6, 0, 0, 1, -9], [0, -5, 0, 0, -35, 4], [0, 3, 0, 0, 9, -2], [0, 4, 0, 0, 6, 7], [-22, 0, 5, -1, 22, -1], [0, 2, 0, 0, -36, -4], [0, 4, 0, 0, -12, -5], [0, -2, 0, 0, -26, -2], [-11, 3, 3, -3, 0, 0], [0, 10, 0, 0, -5, -6], [31, 10, 10, 6, 0, 0], [-17, 4, 4, 9, 0, 0], [0, -10, 0, 0, 0, -8], [0, -8, 0, 0, -9, 13], [15, 0, -4, 3, -15, 3], [18, 0, 6, -2, -18, -2], [-6, 0, -4, -10, 6, -10], [11, 0, -14, -11, -11, -11], [0, 5, 0, 0, -41, 4], [0, 8, 0, 0, 30, -8], [-1, 15, 15, 4, 0, 0], [25, 0, 4, 1, -25, 1], [-30, -9, -9, 1, 0, 0], [-46, 0, -1, 0, 46, 0], [0, 9, 0, 0, -25, -4], [17, 0, -5, -7, -17, -7], [-22, 0, 1, 2, 22, 2], [1, -11, -11, -3, 0, 0], [-23, 0, -4, -6, 23, -6], [27, 0, 9, 7, -27, 7], [23, 0, 2, -6, -23, -6], [25, 0, 6, -7, -25, -7], [-24, -14, -14, -10, 0, 0], [0, -13, 0, 0, -6, 7], [-13, 0, -13, 0, 13, 0], [13, 0, -11, 13, -13, 13], [-4, 0, 9, -11, 4, -11], [0, -16, 0, 0, -22, 9], [0, -6, 0, 0, 13, 2], [-34, 2, 2, 12, 0, 0], [-13, 0, 3, -5, 13, -5], [0, 15, 0, 0, -10, 10], [13, 19, 19, 9, 0, 0], [-29, -17, -17, -7, 0, 0], [0, 8, 0, 0, 13, -6], [0, -2, 0, 0, -26, -8], [24, -5, -5, -16, 0, 0], [4, 0, 4, -2, -4, -2], [0, -3, 0, 0, 49, -3], [0, 0, 0, 0, 18, -6], [-23, 0, 0, -11, 0, 0], [-4, 0, 9, 7, 4, 7], [0, -4, 0, 0, 3, -7], [-4, 2, 2, 9, 0, 0], [17, 2, 2, -3, 0, 0], [0, 13, 0, 0, 13, 0], [5, 0, -1, -18, -5, -18], [-1, 14, 14, 16, 0, 0], [-13, 0, -4, 7, 13, 7], [-22, 10, 10, 22, 0, 0], [0, -1, 0, 0, 10, 6], [-26, 0, 8, 14, 26, 14], [0, -8, 0, 0, 10, 6], [10, 0, 12, 6, -10, 6], [22, 0, -12, -6, -22, -6], [0, 11, 0, 0, -29, -6], [0, -6, 0, 0, 3, 9], [-22, 4, 4, 8, 0, 0], [7, 1, 1, -9, 0, 0], [9, 0, 3, 6, -9, 6], [1, 0, 11, 18, -1, 18], [12, 4, 4, 4, 0, 0], [-11, 1, 1, -10, 0, 0], [0, -9, 0, 0, 34, 11], [0, 11, 0, 0, -13, -5], [23, -10, -10, 5, 0, 0], [-53, 0, -7, -1, 53, -1], [-10, 0, 7, -2, 10, -2], [0, 13, 0, 0, -15, -14], [-5, 0, -2, 9, 5, 9], [22, 0, -11, -18, -22, -18], [-15, 7, 7, -6, 0, 0], [0, -14, -14, -8, 0, 0], [-32, 9, 9, 21, 0, 0], [11, -4, -4, 3, 0, 0], [-20, 0, 2, 8, 20, 8], [25, 9, 9, -4, 0, 0], [-8, 0, 19, -9, 8, -9], [39, 0, 14, 7, -39, 7], [0, 7, 0, 0, -10, 10], [0, 3, 0, 0, 25, 2], [-8, 15, 15, -8, 0, 0], [4, -13, -13, 12, 0, 0], [-7, -2, -2, 21, 0, 0], [0, -12, 0, 0, -28, 6], [2, 0, -7, 10, -2, 10], [0, 6, 0, 0, -8, 0], [0, 3, 0, 0, -26, -10], [-13, 0, -12, 3, 13, 3], [17, -5, -5, 15, 0, 0], [0, 6, 0, 0, -35, 3], [-44, 0, 0, -4, 0, 0], [0, 4, 0, 0, 10, -2], [0, -9, 0, 0, 36, -7], [9, 0, -4, 7, -9, 7], [0, -7, 0, 0, -54, 2], [4, 0, 0, -17, -4, -17], [-8, 0, 21, 17, 8, 17], [-30, 0, 0, 5, 30, 5], [4, 0, -4, 8, -4, 8], [0, -11, 0, 0, 1, 13], [69, -2, -2, 1, 0, 0], [0, 0, 0, 0, -11, 7], [19, 21, 21, 11, 0, 0], [-17, 0, 0, -3, 0, 0], [0, -1, 0, 0, -3, -19], [44, -2, -2, -2, 0, 0], [54, 0, 4, 6, -54, 6], [13, -3, -3, 18, 0, 0], [12, 0, -14, -4, -12, -4], [0, -4, 0, 0, -21, -12], [0, -7, 0, 0, -16, 6], [6, -7, -7, -16, 0, 0], [-29, 0, 15, -9, 29, -9], [-30, 0, -13, -4, 30, -4], [18, 4, 4, 12, 0, 0], [0, -9, 0, 0, -1, 11], [0, 2, 0, 0, 30, -10], [-17, 2, 2, -3, 0, 0], [-7, 2, 2, -5, 0, 0], [-3, -5, -5, 7, 0, 0], [0, -4, 0, 0, -17, 15], [-21, 9, 9, 6, 0, 0], [0, -3, 0, 0, 56, -5], [-19, -11, -11, -7, 0, 0], [14, 0, -6, -2, -14, -2], [0, 8, 0, 0, 11, -4], [0, 0, 0, 0, 26, -3], [-2, 14, 14, 8, 0, 0], [0, -14, 0, 0, -21, -7], [-8, 0, -8, 4, 8, 4], [-7, -1, -1, -19, 0, 0], [-25, 0, -2, -9, 25, -9], [7, 6, 6, -7, 0, 0], [-35, -7, -7, 3, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1110_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1110_2_i_o();".
// 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_i_o(:prec:=6)
    chi := MakeCharacter_1110_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_i_o();".
// 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_i_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1110_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![36, 84, 202, -2, 15, 1, 1]>,<11,R![-2, -11, 0, 1]>],Snew);
    return Vf;
end function;