
// Make newform 1110.2.a.g in Magma, downloaded from the LMFDB on 19 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_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_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
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], [1], [-1], [4], [4], [2], [-2], [0], [0], [-6], [0], [-1], [-6], [4], [12], [-6], [8], [10], [4], [0], [-6], [0], [-4], [-14], [-10], [6], [8], [12], [-14], [14], [12], [-8], [10], [-4], [-10], [0], [10], [4], [8], [2], [-16], [14], [-16], [-2], [2], [-24], [-20], [-20], [-4], [-18], [10], [0], [18], [24], [-18], [-12], [-18], [8], [10], [-14], [-4], [-14], [12], [-16], [6], [2], [-32], [-30], [-12], [30], [-26], [-24], [-20], [2], [-20], [24], [34], [18], [-6], [-30], [20], [-6], [0], [-30], [-24], [-4], [42], [-2], [26], [8], [36], [0], [8], [36], [40], [40], [30], [42], [-36], [26], [-28], [-30], [-44], [2], [-4], [-34], [-12], [18], [32], [26], [24], [10], [2], [-44], [-16], [18], [44], [24], [-30], [-44], [10], [-38], [-30], [-12], [20], [-30], [-14], [-8], [0], [34], [-28], [28], [-32], [10], [-54], [-14], [42], [-12], [34], [-6], [28], [-42], [4], [36], [34], [-24], [10], [-18], [-8], [-12], [26], [-46], [-36], [-44], [4], [40], [16], [50], [2], [-10], [4], [-22], [48], [20], [-18], [-4], [-40], [-54], [50], [42], [24], [50], [-40], [-38], [-32], [2], [32], [-10], [20], [-10], [32], [0], [58], [-42], [-12], [54], [-6], [-4], [18], [8], [-42], [-12], [-8], [-50], [-12], [34], [58], [2], [-30], [56], [50], [8], [34], [-14], [44], [2], [-8], [-44], [2], [-12], [22], [-14], [-40], [36], [32], [42], [-32], [-38], [0], [-54], [30], [-40], [-30], [16], [-20], [-22], [-30], [8], [-52], [-24], [58], [-20], [16], [-38], [36], [36], [34], [10], [72], [48], [-12], [-32], [28], [-14], [-54], [0], [56], [16], [76], [0], [2], [-14], [-36], [-22], [18], [-4], [-2], [-76], [74], [-10], [56], [-36], [70], [2], [-42], [20], [-50], [42], [36], [-70], [-62], [52], [-26], [72], [58], [4], [36], [-14], [-38], [-24], [44], [-16], [-12], [-18], [28], [48], [54], [-62], [0], [66], [-30], [-52], [-46], [-68], [50], [30], [40], [66], [80], [84], [78], [-6], [24], [36], [-16], [-82], [12], [10], [-64], [-6], [-36], [-26], [-30], [76], [-56], [-30], [44], [56], [26], [-30], [8], [2], [-14], [52], [26], [-78], [76], [44], [0], [42], [-34], [-14], [-16], [36], [-80], [52], [70], [2], [50], [-36], [66], [2], [-30], [-24], [66], [24], [22], [28], [-16], [-62], [84], [2], [-6], [72], [-50], [-14], [-16], [40], [-6], [-64], [-22], [74], [72], [-8], [-92], [46], [82], [-16], [-22], [-48], [16], [-76], [6], [-80], [2], [-84], [-48], [90], [26], [26], [18], [14], [-32], [10], [-40], [60], [80], [-54], [60], [48], [82], [-54], [16], [92], [48], [-22], [-16], [90], [52], [46], [-74], [38], [-48], [-26], [-6], [16], [-46], [18], [-20], [28], [-94], [74], [12], [-8], [62], [-26], [56], [-92], [98], [-48], [-62], [-6], [36], [-12], [-14], [42], [-36], [-22], [-60], [88]];
    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_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_a_g(:prec:=1)
    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_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_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1110_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, 1]>,<11,R![-4, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;