
// Make newform 1092.2.a.g in Magma, downloaded from the LMFDB on 23 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1092_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_1092_2_a_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_1092_2_a_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 := [-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_1092_a();"
function MakeCharacter_1092_a()
    N := 1092;
    order := 1;
    char_gens := [547, 365, 157, 925];
    v := [1, 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_1092_a_Hecke(Kf)
    return MakeCharacter_1092_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]];
    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, 1], [1, 0], [2, 0], [-1, 0], [0, 0], [4, -1], [2, -1], [-2, 1], [0, -1], [2, -2], [4, -2], [4, -3], [-4, 1], [2, 1], [-8, 0], [6, 0], [8, -2], [-2, -2], [-2, 5], [4, -3], [4, -3], [-4, -1], [14, 1], [-12, 2], [-8, 4], [-6, 2], [-2, 4], [2, 3], [-4, 0], [-4, 4], [-2, -6], [-4, 2], [2, 0], [8, 4], [-2, -2], [0, -4], [-12, 3], [4, -8], [-2, -5], [2, 0], [-10, 2], [6, 0], [18, 0], [-16, 0], [4, 1], [0, 5], [-16, -4], [2, -4], [-2, 1], [-22, 2], [-2, 1], [0, -2], [-12, 6], [-14, 1], [-24, 0], [0, -8], [2, 3], [22, 0], [-12, -4], [4, 5], [-4, 5], [-12, -2], [10, 0], [10, -2], [4, 6], [2, 7], [-2, 2], [-2, -7], [-8, -2], [-2, 2], [0, -2], [-10, 0], [4, 6], [0, -4], [-2, 0], [-26, 1], [14, 2], [6, 5], [8, -2], [10, -2], [30, -4], [-6, 2], [8, -2], [38, -1], [10, 2], [10, -6], [20, -2], [4, -2], [0, -6], [-28, 3], [16, 2], [14, -2], [-4, 0], [8, -10], [-8, 9], [16, 0], [-4, 10], [-14, 2], [32, -1], [2, 10], [0, 12], [-2, -7], [32, 1], [-6, 8], [24, 1], [16, -11], [6, -5], [18, 4], [-32, 2], [-10, 2], [22, 4], [4, 0], [-12, 14], [6, 3], [28, -4], [12, -2], [-2, -12], [2, 5], [-18, 5], [-10, 11], [28, 6], [-18, 6], [20, 9], [30, -5], [-26, 4], [-32, -6], [-8, -6], [26, -3], [0, 14], [-2, 6], [8, -11], [-2, -9], [-20, 5], [-14, 3], [4, 10], [4, 7], [20, -10], [-46, -1], [-12, -8], [26, -8], [-36, 4], [18, -14], [-38, 0], [4, -4], [22, -13], [12, 6], [-20, 6], [2, 6], [-2, -2], [-40, 2], [-4, 8], [-28, 8], [36, -3], [-2, 5], [-16, 0], [-36, -7], [2, -6], [-24, 3], [26, 8], [18, -1], [-28, 10], [-4, -2], [-2, -8], [0, 11], [24, -4], [6, -2], [-50, 4], [20, 8], [-38, 4], [-2, -8], [38, -11], [38, -6], [8, 8], [-24, 16], [0, 16], [6, 8], [-16, -10], [-2, 8], [-12, 8], [12, 6], [10, 4], [0, -15], [2, -8], [-4, -6], [-50, -3], [28, -9], [6, 8], [-8, 3], [22, 2], [-30, 8], [28, -3], [28, -11], [-14, 17], [-8, -16], [14, 4], [10, 15], [28, 7], [28, -14], [-54, 6], [-24, 16], [18, 9], [6, 6], [-44, -3], [4, 0], [0, -13], [10, -9], [-14, -2], [44, -10], [-22, 9], [8, -12], [32, 8], [-60, 0], [4, 1], [10, -1], [-64, 1], [-8, 8], [34, -6], [46, -2], [-2, 11], [-24, 8], [-46, 2], [0, 18], [28, -6], [-6, 17], [-36, 12], [-22, -3], [-8, -6], [30, 12], [-18, 4], [36, -8], [-8, 4], [10, -12], [-20, 18], [-16, -3], [22, -16], [26, 10], [-50, 9], [-8, -10], [22, -2], [20, 14], [16, 12], [18, -8], [-20, 3], [20, 16], [32, -11], [-56, 7], [-24, -4], [-14, 5], [6, 14], [-40, 6], [-26, 16], [-10, 0], [4, -2], [18, 6], [-14, 9], [28, 0], [44, -4], [42, -7], [20, 13], [6, 21], [42, -11], [6, 20], [48, -2], [-4, 14], [-30, -4], [-32, 15], [32, 10], [30, -20], [-10, -16], [20, -20], [6, 20], [-64, 7], [46, -2], [8, -8], [10, -2], [34, 2], [14, -18], [-8, -20], [24, -6], [36, 10], [16, -4], [-34, -7], [-52, -8], [-34, 15], [-38, -7], [10, -2], [56, 0], [20, 0], [-70, 5], [28, 4], [-10, 8], [24, 3], [22, -21], [36, -12], [-8, -14], [-40, 0], [-32, -16], [-68, 3], [74, -4], [14, 2], [28, 9], [50, -9], [48, 7], [-38, 8], [-26, 6], [-22, 2], [-2, -12], [18, -23], [24, 12], [-2, 12], [-42, 0], [-76, 3], [-30, -6], [22, -22], [-20, -13], [42, -4], [54, -13], [-26, 7], [12, 4], [-10, -18], [-36, -6], [8, 4], [-22, 6], [-34, -10], [-24, -20], [-20, 17], [30, -10], [46, -8], [22, -12], [-42, -3], [32, -22], [56, -7], [48, -2], [-48, -9], [14, 14], [-20, 22], [14, 20], [-48, 6], [38, 2], [62, 8], [0, -14], [-32, 0], [-50, -2], [-22, -18], [-38, 16], [-20, 0], [-56, -5], [4, 2], [-36, -5], [6, 8], [-22, 10], [-8, -10], [54, -8], [8, -10], [-32, 13], [-36, 21], [26, 8], [6, -4], [22, -24], [20, -18], [-2, -17], [-22, -10], [-20, 2], [30, -23], [-16, 5], [-60, 10], [-26, 17], [-2, 14], [-20, 22], [-10, -26], [42, -11], [72, -8], [10, -3], [-12, -16], [18, 6], [-6, -20], [32, -28], [54, 4], [28, 1], [2, 16], [-16, -12], [82, 2], [44, 5], [0, -27], [-16, 4], [-22, -2], [16, 0], [68, -12], [-18, 24], [32, -25], [2, -22], [-60, 16], [62, -10], [10, 7], [24, 0], [56, -18], [-56, -13], [62, 2], [14, -5], [-78, -7], [-4, -2], [76, 7], [64, 5], [8, -20], [2, 14], [-86, 6], [-68, 13], [-24, 13], [34, -18], [-26, -1], [62, -15], [4, -16], [30, -15], [-80, 8], [-14, 26], [-48, 8], [-48, 2], [20, 16], [78, -3], [22, -16], [-18, -22], [-12, 24], [26, 16], [36, 17], [-26, 7], [38, -18], [-28, 13], [-12, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1092_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1092_2_a_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_1092_2_a_g(:prec:=2)
    chi := MakeCharacter_1092_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_1092_2_a_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_1092_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1092_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-8, -1, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;