
// Make newform 1110.2.a.o in Magma, downloaded from the LMFDB on 20 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_o();".
// 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_o();".
// 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], [-2], [2], [-2], [2], [0], [2], [4], [1], [-6], [-4], [-10], [6], [4], [4], [-12], [8], [-2], [-4], [4], [-6], [12], [0], [6], [-8], [-16], [-6], [-12], [4], [12], [-20], [4], [-8], [-2], [-4], [-20], [-10], [-12], [-18], [0], [12], [-6], [16], [16], [20], [-12], [-22], [8], [16], [10], [0], [-6], [26], [-24], [16], [-2], [6], [20], [-14], [12], [8], [-4], [-18], [-10], [14], [-12], [2], [18], [-24], [28], [10], [4], [20], [-18], [-6], [30], [38], [6], [-20], [8], [30], [-8], [36], [-30], [-20], [14], [6], [-20], [40], [42], [-22], [22], [24], [24], [-18], [20], [-4], [16], [-34], [4], [22], [32], [8], [28], [44], [-4], [-26], [38], [-30], [28], [0], [-48], [34], [8], [8], [14], [30], [-8], [2], [-14], [12], [-40], [30], [-8], [-20], [6], [-46], [-32], [-6], [0], [30], [-50], [26], [54], [28], [30], [10], [20], [16], [-56], [-4], [-48], [48], [-26], [10], [14], [-26], [-14], [14], [32], [-22], [28], [-40], [24], [-10], [14], [8], [12], [-36], [-46], [-6], [-10], [-14], [4], [-30], [6], [-6], [-36], [-12], [-48], [-18], [24], [54], [-22], [28], [-16], [-62], [-62], [48], [-10], [18], [-38], [44], [-34], [0], [-26], [-64], [16], [64], [26], [12], [32], [-56], [14], [2], [4], [0], [-6], [-8], [10], [14], [-30], [-50], [8], [60], [-54], [52], [-24], [30], [38], [-24], [64], [-2], [-2], [30], [-52], [-54], [2], [64], [42], [-14], [-28], [-36], [-72], [-20], [28], [-52], [50], [8], [40], [-2], [-44], [70], [62], [-6], [-12], [-40], [4], [-2], [36], [24], [68], [-56], [2], [12], [-32], [36], [-2], [-6], [-2], [10], [-18], [-30], [6], [-20], [54], [20], [18], [36], [22], [34], [70], [-8], [12], [-42], [-12], [26], [4], [-56], [28], [-56], [2], [44], [48], [40], [-74], [-52], [34], [52], [38], [-62], [8], [-52], [-76], [-30], [52], [-30], [-46], [-76], [84], [50], [-50], [-28], [-64], [-46], [0], [44], [56], [26], [24], [28], [-58], [32], [-12], [40], [40], [-62], [2], [16], [22], [52], [-48], [-66], [86], [0], [18], [46], [70], [90], [30], [-76], [24], [-14], [-16], [-40], [-32], [14], [30], [-34], [-60], [-4], [2], [-76], [-30], [-24], [-10], [-84], [94], [52], [74], [32], [-22], [-48], [-10], [-44], [64], [-78], [-76], [-18], [-70], [66], [-34], [-12], [-48], [44], [-48], [-56], [-46], [-6], [84], [44], [-84], [4], [30], [66], [-78], [84], [-70], [10], [-64], [52], [74], [-18], [-24], [62], [-30], [2], [58], [54], [74], [-20], [-14], [54], [48], [62], [32], [68], [-50], [42], [-48], [-32], [-12], [-6], [16], [18], [52], [56], [-70], [94], [-42], [18], [-26], [56], [102], [38], [-36], [90], [-22], [6], [4], [54], [-96], [-36], [12], [12], [-4], [-48], [10], [90], [18], [-14], [-46], [18], [84], [-30], [60], [-24]];
    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_o();".
// 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_o(: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_o();".
// 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_o( : 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![2, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;