
// Make newform 1110.2.a.p in Magma, downloaded from the LMFDB on 18 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_p();".
// 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_p();".
// 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, -1, 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_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], [-1, 0], [1, 0], [0, -1], [2, -1], [-2, 1], [2, -1], [2, -1], [0, 1], [6, 0], [-4, 2], [-1, 0], [2, 2], [-4, 0], [2, -2], [2, 1], [8, 2], [4, -2], [-4, 2], [8, -2], [6, -3], [4, 0], [4, -1], [10, 1], [0, 4], [4, 2], [6, -2], [0, 1], [8, -1], [6, 4], [0, 1], [0, 0], [4, 2], [-4, 2], [16, -2], [0, -1], [10, -2], [-12, 5], [-12, 3], [10, -5], [-8, 4], [6, -8], [-16, -1], [-8, 2], [-2, -5], [8, 0], [-16, 2], [-8, 8], [4, 0], [-6, 0], [-16, -2], [8, -4], [-14, 4], [-20, -2], [-2, -1], [-10, 6], [4, -1], [0, 4], [2, 7], [-2, 1], [12, -7], [6, -3], [-4, -2], [-8, 0], [8, 0], [10, -8], [-2, -2], [6, -3], [-12, 4], [10, 2], [-10, 4], [-8, 12], [24, -1], [-10, 4], [4, 6], [4, 7], [10, 4], [14, 2], [-26, -3], [-2, -2], [18, -3], [4, 6], [-8, -7], [6, -5], [-32, 0], [-4, 0], [10, 8], [0, -12], [-6, 4], [14, -10], [-12, 4], [-8, -3], [10, -12], [6, -1], [-10, 9], [-8, 4], [-12, 9], [-26, 6], [20, -8], [-12, 5], [8, -3], [34, 2], [20, -2], [-2, -9], [-8, 2], [20, 2], [-12, 4], [4, -8], [4, -8], [14, 5], [-2, 12], [-10, 12], [12, 6], [0, -8], [40, 0], [10, 10], [0, 7], [-16, -7], [-14, 10], [18, -6], [-16, -3], [-14, 5], [14, -5], [4, 8], [16, 2], [-14, -10], [-24, 1], [20, -10], [-18, 8], [-10, 4], [-8, 10], [-18, 12], [-24, 4], [34, 3], [-26, 4], [-38, -4], [10, 3], [4, 2], [18, -4], [-14, -3], [20, 0], [-4, 9], [36, -3], [-20, 0], [8, -13], [-16, 2], [10, 9], [-26, 13], [-26, 13], [10, -2], [-34, -6], [6, -6], [-40, -3], [-42, 0], [20, -7], [8, 8], [-48, 4], [-26, 4], [22, 0], [36, -6], [12, -12], [-44, -2], [10, -8], [30, 6], [42, -5], [-26, 4], [-4, 8], [6, 3], [-26, 8], [-10, 2], [32, 2], [12, 2], [8, 8], [-42, -3], [-8, -13], [6, -12], [-6, -2], [16, -11], [36, 4], [2, 16], [-30, -2], [12, -6], [-30, 1], [-50, 4], [-2, 10], [32, 3], [26, -6], [24, 11], [-58, 2], [-16, 12], [-28, 14], [0, 4], [-46, 3], [16, 11], [-16, -4], [8, -8], [-42, 6], [6, -8], [60, 0], [40, -9], [54, -4], [40, -5], [-10, -2], [-34, 12], [30, -2], [18, -12], [24, -1], [44, 1], [-54, 1], [-4, 8], [12, -10], [10, 10], [38, -2], [0, 9], [-24, -1], [-58, -1], [-34, -4], [14, -24], [-36, 1], [46, -5], [34, -8], [-8, 7], [26, 0], [34, -8], [-20, -8], [-4, -15], [24, 12], [52, 4], [-24, 20], [-32, 8], [30, -8], [-8, 24], [32, -3], [30, -18], [36, 8], [26, -8], [22, -9], [6, 0], [40, -2], [-24, 0], [-28, -4], [-2, -13], [-40, 9], [-16, -7], [36, -12], [0, -1], [10, 0], [0, 2], [-24, -10], [-4, 15], [2, 1], [-14, -2], [2, -4], [18, -10], [-46, 0], [22, -14], [-34, 16], [-20, -14], [34, -3], [0, -10], [-6, 8], [-28, 8], [-10, 6], [6, -20], [34, -13], [0, -16], [16, 7], [14, -4], [20, 6], [6, -4], [4, -15], [-16, 9], [24, 4], [32, 0], [2, 7], [-8, -11], [-8, -9], [-56, 6], [-2, -19], [-8, 10], [14, 16], [12, 6], [-30, -4], [50, -6], [40, 12], [-28, -10], [-24, 10], [38, 0], [28, 6], [-22, -7], [-50, 4], [44, -2], [20, -18], [38, 1], [34, 10], [-16, 22], [48, -1], [-50, 13], [-12, 12], [-44, -4], [-52, -2], [-50, 4], [8, 24], [-12, -6], [-26, 1], [-4, -4], [60, -12], [-80, -2], [-24, -10], [-10, -15], [-66, 6], [-52, 1], [54, 0], [-12, -12], [-40, -1], [6, -4], [58, -11], [8, -3], [58, 0], [62, -1], [6, -7], [-38, -3], [18, 10], [-40, -5], [-32, 0], [-54, -2], [16, 0], [-8, 16], [-40, -13], [-38, 0], [14, -10], [34, 2], [28, 6], [-28, 6], [-22, -7], [44, 4], [-14, -12], [0, -10], [-10, 12], [-48, 0], [-70, 2], [20, -22], [30, 6], [-24, 16], [-42, 6], [-28, -14], [-10, 10], [-52, -6], [88, 0], [6, 4], [4, -18], [14, 3], [-34, 10], [-54, 4], [-42, -8], [20, 8], [0, 11], [-16, -14], [-16, -16], [0, -24], [-2, -24], [-54, -12], [-28, 3], [32, 0], [-44, -2], [8, 18], [50, 2], [-38, 4], [2, 28], [-8, -4], [58, 6], [-2, 16], [-20, 7], [-20, -6], [-18, 16], [50, -1], [8, 18], [14, -15], [18, -9], [-6, -23], [-10, -4], [18, -21], [2, 0], [44, -10], [-30, 10], [-38, 18], [16, -9], [-14, 8], [-32, -12], [-4, -1], [-10, -17], [-74, 12], [-28, -12], [-8, -7], [60, -12], [-30, -11], [56, 6], [-6, -22], [-4, 2], [60, -11], [26, 2], [26, 3], [-10, 26], [-18, 8], [-22, -16], [-8, -9], [-14, 11], [-50, 6], [28, -12], [-18, 30], [82, 3], [-14, 1], [-36, 18], [22, -11], [20, -28], [-8, -11], [52, 12], [8, 1], [-44, 14], [8, -16], [86, -8], [30, -11], [-10, -12], [-18, 15], [-14, -11], [14, 8], [-44, -11], [10, -12], [36, -10], [-64, 4]];
    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_p();".
// 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_p(:prec:=2)
    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_p();".
// 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_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1110_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-8, 1, 1]>,<11,R![-6, -3, 1]>,<13,R![-6, 3, 1]>],Snew);
    return Vf;
end function;