
// Make newform 1110.2.d.i in Magma, downloaded from the LMFDB on 25 April 2024.
// 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_i();".
// 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_i();".
// 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 := [8, -20, 25, -4, 0, 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], [100, -121, 10, -25, -2, -5], [223, 0, -14, 35, 27, 7], [258, -605, 50, -4, -10, -25], [574, -1331, 372, 38, -26, -65]];
    Rf_basisdens := [1, 1, 121, 121, 242, 242];
    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_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)
    N := 1110;
    order := 2;
    char_gens := [371, 667, 631];
    char_values := [[1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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], [0, 0, 0, 0, -1, 0], [0, -1, 0, -1, 0, 0], [0, -1, 1, 0, 1, 0], [-1, 1, 1, 2, 0, 0], [0, 0, 0, 0, 1, 1], [0, 1, -1, 0, -3, -2], [-3, 1, 1, 1, 0, 0], [0, -1, 1, 0, 3, -1], [6, 2, 2, 1, 0, 0], [0, -1, -1, 1, 0, 0], [0, 0, 0, 0, -1, 0], [-2, 0, 0, 3, 0, 0], [0, -1, 1, 0, 0, -3], [0, 1, -1, 0, -2, -2], [0, 0, 0, 0, 3, -2], [2, 3, 3, 2, 0, 0], [0, -3, -3, -3, 0, 0], [0, 0, 0, 0, -10, -2], [0, 3, 3, -2, 0, 0], [0, 1, -1, 0, -1, 1], [4, 1, 1, 0, 0, 0], [0, -2, 2, 0, 7, 3], [-1, 0, 0, 3, 0, 0], [0, -2, 2, 0, 0, 7], [6, 4, 4, 4, 0, 0], [0, 1, -1, 0, 12, 4], [0, -2, 2, 0, -7, -5], [-3, -1, -1, -4, 0, 0], [0, 1, -1, 0, -2, -5], [0, 2, -2, 0, -5, 3], [4, 5, 5, 6, 0, 0], [0, 1, -1, 0, -14, 0], [8, -3, -3, 1, 0, 0], [-4, -6, -6, 0, 0, 0], [15, -2, -2, 1, 0, 0], [0, -5, 5, 0, -2, 7], [0, 0, 0, 0, 7, 0], [0, -5, 5, 0, -7, 3], [0, -2, 2, 0, -1, 2], [-4, 0, 0, 6, 0, 0], [10, 3, 3, 2, 0, 0], [5, 2, 2, 0, 0, 0], [0, 0, 0, 0, -6, 0], [0, 1, -1, 0, -3, -9], [-16, -4, -4, 0, 0, 0], [6, 8, 8, 3, 0, 0], [0, -3, 3, 0, -10, 1], [0, -9, 9, 0, 2, 3], [-6, 3, 3, -6, 0, 0], [0, 2, -2, 0, 2, 0], [8, 7, 7, 1, 0, 0], [-12, -7, -7, -6, 0, 0], [-2, -4, -4, -10, 0, 0], [0, -2, 2, 0, 3, -7], [0, -4, 4, 0, -4, 7], [5, 1, 1, 5, 0, 0], [16, -4, -4, 6, 0, 0], [0, -4, 4, 0, -7, 1], [-15, -2, -2, -5, 0, 0], [0, -3, 3, 0, -7, -3], [0, 6, -6, 0, 9, -6], [0, -1, 1, 0, -4, 4], [-16, 3, 3, -5, 0, 0], [0, 1, -1, 0, -2, 0], [0, 1, -1, 0, 14, 7], [2, -6, -6, -4, 0, 0], [0, -7, 7, 0, 11, 3], [0, 5, -5, 0, -4, -4], [4, -5, -5, -6, 0, 0], [0, -3, 3, 0, 12, 11], [-6, -3, -3, -8, 0, 0], [0, 3, -3, 0, -3, 0], [0, -3, 3, 0, -2, 0], [-16, 7, 7, 8, 0, 0], [0, 3, -3, 0, 7, -9], [8, 3, 3, 5, 0, 0], [0, 8, -8, 0, 4, -10], [5, -6, -6, -9, 0, 0], [-6, -7, -7, -6, 0, 0], [1, 4, 4, 5, 0, 0], [16, 0, 0, -6, 0, 0], [-3, -12, -12, -8, 0, 0], [0, 1, -1, 0, 17, -3], [-2, -8, -8, -5, 0, 0], [0, 8, -8, 0, -6, 0], [2, 0, 0, -10, 0, 0], [0, 1, -1, 0, 6, 13], [-24, -1, -1, -1, 0, 0], [0, -3, 3, 0, 12, -4], [0, 0, 0, 0, -18, 5], [-9, -3, -3, -1, 0, 0], [0, 2, -2, 0, -10, -2], [7, 0, 0, 11, 0, 0], [5, 11, 11, 11, 0, 0], [0, 4, -4, 0, -2, -8], [9, -5, -5, -13, 0, 0], [-6, 0, 0, -15, 0, 0], [0, -2, 2, 0, -6, 6], [-1, 2, 2, 1, 0, 0], [0, 10, -10, 0, 9, -8], [0, 9, -9, 0, -10, -12], [0, 8, -8, 0, 4, -13], [7, -8, -8, -19, 0, 0], [14, 6, 6, 3, 0, 0], [0, -2, 2, 0, 6, -4], [0, -9, 9, 0, 6, 13], [0, 0, 0, 0, -24, -4], [-10, -11, -11, -2, 0, 0], [1, 4, 4, 8, 0, 0], [0, 10, -10, 0, -14, -4], [0, -7, 7, 0, 12, 1], [0, 5, -5, 0, 14, -12], [-4, 8, 8, -3, 0, 0], [6, 11, 11, 11, 0, 0], [-26, 3, 3, -13, 0, 0], [0, 2, -2, 0, 5, 2], [0, 3, -3, 0, 17, 3], [0, 4, -4, 0, 12, -2], [20, -2, -2, 4, 0, 0], [-5, -7, -7, -10, 0, 0], [0, 1, -1, 0, 23, 3], [0, 5, -5, 0, -7, -17], [0, -4, 4, 0, 2, -7], [-10, 10, 10, 3, 0, 0], [12, 10, 10, 6, 0, 0], [17, -5, -5, -8, 0, 0], [16, 9, 9, -4, 0, 0], [0, 1, -1, 0, -2, -8], [0, -7, 7, 0, 0, 13], [0, 7, 7, 17, 0, 0], [0, 2, -2, 0, -38, 1], [-4, 6, 6, 12, 0, 0], [0, 6, -6, 0, -21, -5], [-48, -1, -1, -3, 0, 0], [-4, -9, -9, -4, 0, 0], [0, 2, -2, 0, 13, -6], [0, -2, 2, 0, -18, 8], [0, -15, 15, 0, 12, 8], [15, -6, -6, -13, 0, 0], [-20, -8, -8, -12, 0, 0], [3, -3, -3, -11, 0, 0], [0, -16, 16, 0, 5, 5], [0, 14, -14, 0, -6, -5], [13, -1, -1, 8, 0, 0], [24, -1, -1, 12, 0, 0], [0, -12, 12, 0, -13, 9], [0, 9, -9, 0, 15, 10], [23, -5, -5, 11, 0, 0], [0, -4, 4, 0, 20, 9], [0, -14, 14, 0, -2, 7], [14, -7, -7, 11, 0, 0], [0, 10, -10, 0, 17, 0], [0, -1, 1, 0, 36, 5], [0, -9, 9, 0, 23, 7], [32, 0, 0, 12, 0, 0], [12, -3, -3, -8, 0, 0], [20, -7, -7, -15, 0, 0], [0, 4, -4, 0, 24, 0], [-12, 10, 10, 14, 0, 0], [0, 2, -2, 0, -14, -13], [0, 9, -9, 0, 18, -8], [0, -17, 17, 0, 2, 8], [4, -5, -5, 13, 0, 0], [0, -9, 9, 0, -7, 6], [0, -13, 13, 0, -8, -3], [6, -1, -1, -15, 0, 0], [0, 2, -2, 0, 9, 13], [6, 1, 1, 15, 0, 0], [0, 0, 0, 0, -4, -10], [10, -5, -5, -2, 0, 0], [18, 11, 11, 3, 0, 0], [-22, -9, -9, -11, 0, 0], [0, -1, 1, 0, -15, 5], [-7, -20, -20, -7, 0, 0], [16, -6, -6, -4, 0, 0], [4, -14, -14, -18, 0, 0], [5, 13, 13, 5, 0, 0], [0, -3, 3, 0, -24, -23], [-6, 7, 7, 20, 0, 0], [0, 12, -12, 0, 0, -16], [26, -5, -5, 8, 0, 0], [0, 4, -4, 0, -5, -9], [0, 5, -5, 0, 28, 15], [0, -4, 4, 0, 8, 21], [11, 5, 5, -9, 0, 0], [0, -12, 12, 0, 26, 9], [0, 2, -2, 0, 29, 6], [2, 11, 11, 0, 0, 0], [16, 6, 6, -6, 0, 0], [0, 14, -14, 0, -22, -16], [0, 4, -4, 0, -20, -2], [25, 5, 5, 3, 0, 0], [-9, -3, -3, -3, 0, 0], [0, 0, 0, 0, 20, 16], [0, -20, 20, 0, 0, 14], [4, 0, 0, 8, 0, 0], [0, 0, 0, 0, 2, -8], [0, -13, 13, 0, -8, 10], [0, 11, -11, 0, -33, -13], [-42, 4, 4, -8, 0, 0], [-7, -12, -12, -15, 0, 0], [0, -11, 11, 0, 6, 3], [-18, 1, 1, -17, 0, 0], [-2, -9, -9, 9, 0, 0], [0, -10, 10, 0, -10, 16], [31, -4, -4, 1, 0, 0], [0, 0, 0, 0, -11, 5], [13, 2, 2, 13, 0, 0], [-20, 8, 8, 21, 0, 0], [0, 9, -9, 0, 28, 7], [-30, -11, -11, -1, 0, 0], [0, 6, -6, 0, 12, 0], [0, 6, -6, 0, 37, -11], [29, 0, 0, 0, 0, 0], [21, -2, -2, -14, 0, 0], [0, 12, -12, 0, -10, 4], [-6, -22, -22, -16, 0, 0], [0, 1, -1, 0, 23, -5], [0, 12, -12, 0, -11, -18], [6, -4, -4, -12, 0, 0], [-29, 8, 8, -9, 0, 0], [32, 7, 7, 11, 0, 0], [0, 6, -6, 0, 50, -2], [0, 4, -4, 0, -16, -26], [-15, 16, 16, -1, 0, 0], [0, -3, 3, 0, -40, -16], [0, 16, 16, 0, 0, 0], [0, 7, -7, 0, 14, -7], [12, -6, -6, -8, 0, 0], [0, -23, 23, 0, 16, 7], [-22, -21, -21, -14, 0, 0], [67, -2, -2, -3, 0, 0], [-12, 1, 1, -6, 0, 0], [0, 25, -25, 0, -10, -20], [0, -3, 3, 0, 36, 1], [-17, 5, 5, 17, 0, 0], [0, -9, 9, 0, -24, -16], [-22, 16, 16, 18, 0, 0], [-18, -6, -6, -18, 0, 0], [0, -1, 1, 0, 6, 27], [13, 17, 17, -1, 0, 0], [0, 16, -16, 0, -1, -9], [-31, -3, -3, -18, 0, 0], [0, -5, 5, 0, 8, -4], [27, 10, 10, 16, 0, 0], [0, 4, -4, 0, -18, 6], [2, -2, -2, -16, 0, 0], [-4, 13, 13, 3, 0, 0], [0, -7, 7, 0, 1, 13], [0, 16, -16, 0, -19, -17], [-8, 17, 17, 14, 0, 0], [0, 7, -7, 0, 4, -25], [18, -11, -11, -8, 0, 0], [0, -9, 9, 0, 12, -8], [-16, -3, -3, -5, 0, 0], [6, 12, 12, 6, 0, 0], [0, -7, 7, 0, -28, -10], [0, 2, -2, 0, 25, -6], [0, -16, 16, 0, -22, 23], [0, 1, -1, 0, 18, 16], [0, -12, 12, 0, 40, 16], [4, 6, 6, 18, 0, 0], [0, -20, 20, 0, -4, 3], [0, 0, 0, 0, 3, 25], [6, 11, 11, -3, 0, 0], [-35, -17, -17, -9, 0, 0], [10, 16, 16, 22, 0, 0], [0, 9, -9, 0, 44, -2], [0, 8, -8, 0, -38, -14], [-11, 10, 10, 11, 0, 0], [0, 0, 0, 0, 21, -10], [0, -1, 1, 0, 10, -12], [2, 11, 11, 29, 0, 0], [0, -13, 13, 0, -3, 15], [0, 9, -9, 0, -3, -12], [0, 18, -18, 0, -19, -15], [-18, -16, -16, -12, 0, 0], [-39, 2, 2, 12, 0, 0], [46, 7, 7, 10, 0, 0], [0, 8, -8, 0, -30, -14], [-28, 0, 0, -5, 0, 0], [0, 13, -13, 0, -4, 13], [8, 0, 0, 2, 0, 0], [0, 16, -16, 0, -42, -2], [-52, -1, -1, -4, 0, 0], [0, -8, 8, 0, -14, 7], [0, -21, 21, 0, 22, 19], [-42, -21, -21, -6, 0, 0], [-1, -4, -4, 21, 0, 0], [-14, 0, 0, -7, 0, 0], [0, -21, 21, 0, -18, 38], [0, 15, -15, 0, 26, -2], [13, -19, -19, 0, 0, 0], [0, -8, 8, 0, 22, 1], [32, -4, -4, 18, 0, 0], [15, -4, -4, -21, 0, 0], [0, 19, -19, 0, -3, -29], [-24, -8, -8, -32, 0, 0], [0, 14, -14, 0, 24, 2], [0, 5, -5, 0, 24, 5], [0, -8, 8, 0, -50, 0], [-42, -20, -20, -8, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1110_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1110_2_d_i();".
// 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_i(:prec:=6)
    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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1110_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![289, 0, 151, 0, 23, 0, 1]>,<11,R![-59, -11, 5, 1]>],Snew);
    return Vf;
end function;