
// Make newform 1110.2.d.g in Magma, downloaded from the LMFDB on 24 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1110_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1110_d_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_d_g();".
// 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_d_g();".
// 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, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_d();"
function MakeCharacter_1110_d()
    N := 1110;
    order := 2;
    char_gens := [371, 667, 631];
    v := [2, 1, 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_d_Hecke();"
function MakeCharacter_1110_d_Hecke( : Kf := false)
    N := 1110;
    order := 2;
    char_gens := [371, 667, 631];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 1, 0], [0, 0, -1, 0], [0, -1, 0, -2], [0, 1, 1, 1], [-5, -1, 0, 1], [0, 3, 1, 3], [0, 1, -5, 1], [5, -2, 0, 2], [0, 0, 3, 0], [0, -5, 0, 5], [2, -4, 0, 4], [0, 0, 1, 0], [-8, 1, 0, -1], [0, 6, 2, 6], [0, 7, 2, 7], [0, -2, -1, -2], [6, 5, 0, -5], [2, -6, 0, 6], [0, 2, -2, 2], [-8, -1, 0, 1], [0, -2, 5, -2], [-8, -3, 0, 3], [0, 7, -3, 7], [7, -7, 0, 7], [0, 9, -6, 9], [6, -8, 0, 8], [0, -5, -8, -5], [0, -5, -9, -5], [-9, 3, 0, -3], [0, -2, 0, -2], [0, 1, -11, 1], [-8, 7, 0, -7], [0, -11, -2, -11], [-6, 6, 0, -6], [-12, 6, 0, -6], [1, 1, 0, -1], [0, 0, 0, 0], [0, 10, 1, 10], [0, 4, 13, 4], [0, 4, -5, 4], [16, 2, 0, -2], [22, -1, 0, 1], [3, 4, 0, -4], [0, -8, 2, -8], [0, -10, -1, -10], [-20, 0, 0, 0], [-8, 13, 0, -13], [0, 4, 4, 4], [0, 0, -12, 0], [-6, -9, 0, 9], [0, -6, 6, -6], [2, -16, 0, 16], [4, -1, 0, 1], [2, 2, 0, -2], [0, -1, 11, -1], [0, 1, 30, 1], [3, 16, 0, -16], [-4, -2, 0, 2], [0, 7, -7, 7], [5, -13, 0, 13], [0, -14, 1, -14], [0, -4, -3, -4], [0, 7, 4, 7], [-10, -2, 0, 2], [0, -9, 10, -9], [0, 2, -24, 2], [-2, 18, 0, -18], [0, -12, -11, -12], [0, 11, 8, 11], [-4, -17, 0, 17], [0, 10, 2, 10], [-22, -5, 0, 5], [0, -3, -11, -3], [0, -3, 6, -3], [16, 9, 0, -9], [0, -4, -9, -4], [6, 12, 0, -12], [0, 6, 0, 6], [9, -17, 0, 17], [-14, -13, 0, 13], [15, 9, 0, -9], [12, -18, 0, 18], [27, 2, 0, -2], [0, 6, 15, 6], [-12, -9, 0, 9], [0, 4, 18, 4], [-6, 14, 0, -14], [0, -12, 16, -12], [18, 2, 0, -2], [0, 7, 20, 7], [0, -3, -36, -3], [-9, 2, 0, -2], [0, 0, -30, 0], [-15, -1, 0, 1], [1, -6, 0, 6], [0, 4, 18, 4], [19, 8, 0, -8], [8, -17, 0, 17], [0, 0, 34, 0], [15, 15, 0, -15], [0, -16, 23, -16], [0, 11, 22, 11], [0, -1, 18, -1], [-13, -17, 0, 17], [-4, -17, 0, 17], [0, 6, -18, 6], [0, 2, 0, 2], [0, 12, -12, 12], [-6, -19, 0, 19], [-7, 12, 0, -12], [0, -10, 26, -10], [0, 8, -10, 8], [0, 21, 6, 21], [-14, 15, 0, -15], [16, -6, 0, 6], [-24, -2, 0, 2], [0, -18, 7, -18], [0, 0, 21, 0], [0, 6, 12, 6], [-20, -22, 0, 22], [21, 3, 0, -3], [0, -12, 17, -12], [0, 14, -21, 14], [0, 5, -24, 5], [32, -1, 0, 1], [-8, 16, 0, -16], [-37, -5, 0, 5], [0, -1, 0, 1], [0, -13, -2, -13], [0, -12, 6, -12], [10, 0, 0, 0], [0, 25, 16, 25], [-4, -22, 0, 22], [0, 15, -5, 15], [14, 8, 0, -8], [4, 11, 0, -11], [0, 4, 9, 4], [0, 6, 30, 6], [0, 19, 24, 19], [-33, 15, 0, -15], [0, 0, 0, 0], [-23, -20, 0, 20], [0, 9, -5, 9], [0, 5, 0, 5], [15, 3, 0, -3], [0, -15, 0, 15], [0, 31, -1, 31], [0, -35, -7, -35], [15, 6, 0, -6], [0, 11, 10, 11], [0, 3, 36, 3], [-4, 16, 0, -16], [0, -12, 39, -12], [0, 10, 6, 10], [0, 14, -17, 14], [0, 0, 0, 0], [-12, -3, 0, 3], [6, -6, 0, 6], [0, 24, 16, 24], [-32, -4, 0, 4], [0, 17, 4, 17], [0, 1, -2, 1], [0, -7, -50, -7], [-6, 12, 0, -12], [0, -21, -9, -21], [0, 22, -6, 22], [-8, 16, 0, -16], [0, -15, -31, -15], [-8, 0, 0, 0], [0, -22, -24, -22], [30, 13, 0, -13], [20, -14, 0, 14], [8, -16, 0, 16], [0, -12, 15, -12], [27, -9, 0, 9], [-8, 22, 0, -22], [8, 32, 0, -32], [-5, 8, 0, -8], [0, 24, 6, 24], [-30, 9, 0, -9], [0, 0, 0, 0], [-42, -9, 0, 9], [0, 21, -21, 21], [0, -14, 34, -14], [0, -21, -6, -21], [-3, 6, 0, -6], [0, -17, -4, -17], [0, -2, -17, -2], [34, 3, 0, -3], [-24, 16, 0, -16], [0, -34, -2, -34], [0, -2, 40, -2], [-35, 12, 0, -12], [13, -28, 0, 28], [0, 20, 20, 20], [0, -14, 24, -14], [28, -4, 0, 4], [0, -20, 22, -20], [0, 1, -4, 1], [0, -8, -1, -8], [-18, -12, 0, 12], [19, 15, 0, -15], [0, -10, 40, -10], [0, 0, 0, 0], [-4, -32, 0, 32], [0, 6, -6, 6], [-7, -9, 0, 9], [0, 47, -1, 47], [45, -15, 0, 15], [-34, 3, 0, -3], [0, -6, 30, -6], [-12, 0, 0, 0], [0, 14, 8, 14], [0, -11, -25, -11], [3, -14, 0, 14], [17, 8, 0, -8], [0, 0, 6, 0], [6, -2, 0, 2], [0, -22, 23, -22], [0, 14, 9, 14], [-10, 8, 0, -8], [33, 21, 0, -21], [-14, -22, 0, 22], [0, -4, -34, -4], [0, -30, 12, -30], [25, 11, 0, -11], [0, 1, 44, 1], [-36, -12, 0, 12], [0, -10, 44, -10], [-24, -30, 0, 30], [0, 26, 26, 26], [2, 39, 0, -39], [-51, -3, 0, 3], [48, 17, 0, -17], [0, -11, -10, -11], [0, -32, 30, -32], [-19, -2, 0, 2], [0, -19, 12, -19], [30, -18, 0, 18], [-2, -8, 0, 8], [0, 16, 0, 16], [17, -8, 0, 8], [0, -9, 1, -9], [15, 39, 0, -39], [0, 15, 12, 15], [-3, 4, 0, -4], [0, 18, 42, 18], [-18, -18, 0, 18], [-58, -8, 0, 8], [0, -44, 9, -44], [0, 33, 9, 33], [-16, 25, 0, -25], [0, 8, -66, 8], [6, 9, 0, -9], [0, -15, -36, -15], [-14, -28, 0, 28], [-18, 6, 0, -6], [0, 3, -20, 3], [0, 24, -3, 24], [0, -21, -8, -21], [0, -9, -46, -9], [0, 0, -48, 0], [-20, 0, 0, 0], [0, -39, 6, -39], [0, -35, 27, -35], [28, -20, 0, 20], [23, -28, 0, 28], [-6, 50, 0, -50], [0, 7, 20, 7], [0, -2, -14, -2], [21, 9, 0, -9], [0, -12, -1, -12], [0, 41, -2, 41], [8, 48, 0, -48], [0, -26, 31, -26], [0, -37, 13, -37], [0, -19, -57, -19], [2, 16, 0, -16], [33, 6, 0, -6], [-2, 25, 0, -25], [0, -22, 46, -22], [-22, 15, 0, -15], [0, -12, 30, -12], [16, 30, 0, -30], [0, -30, -18, -30], [16, -35, 0, 35], [0, 23, 20, 23], [0, 28, 24, 28], [10, -3, 0, 3], [3, -41, 0, 41], [12, -3, 0, 3], [0, 3, -18, 3], [0, 21, 18, 21], [-35, 1, 0, -1], [0, 3, 24, 3], [-36, -34, 0, 34], [-59, -11, 0, 11], [0, 16, 47, 16], [0, 24, 0, -24], [0, -24, -8, -24], [0, -24, -6, -24], [0, 8, -18, 8], [46, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1110_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1110_2_d_g();".
// 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_d_g(:prec:=4)
    chi := MakeCharacter_1110_d();
    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_d_g();".
// 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_d_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1110_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 0, 6, 0, 1]>,<11,R![23, 10, 1]>],Snew);
    return Vf;
end function;