
// Make newform 1110.2.a.l in Magma, downloaded from the LMFDB on 23 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_l();".
// 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_l();".
// 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], [3], [1], [1], [-1], [-5], [7], [-2], [2], [1], [4], [-12], [12], [-9], [10], [14], [2], [2], [-9], [4], [-1], [1], [-8], [6], [8], [-19], [-19], [-2], [13], [0], [-10], [-6], [6], [-17], [-8], [-9], [-7], [-23], [12], [2], [5], [-22], [13], [4], [-10], [0], [-4], [26], [-14], [20], [-14], [18], [-13], [8], [-5], [-28], [-17], [-11], [-21], [-9], [18], [24], [8], [-6], [12], [-1], [24], [-4], [14], [4], [35], [10], [-2], [13], [14], [-16], [25], [-8], [-1], [22], [3], [-19], [24], [-36], [-18], [24], [6], [-36], [-12], [-9], [6], [-35], [-11], [36], [-11], [36], [-20], [3], [-13], [-12], [-6], [-1], [18], [22], [-12], [-8], [36], [-13], [-14], [-2], [-42], [-4], [32], [44], [-7], [-17], [24], [-36], [7], [19], [-3], [40], [-2], [-40], [27], [10], [30], [26], [54], [14], [44], [43], [-14], [38], [9], [14], [-46], [5], [-16], [-3], [13], [36], [-7], [-54], [21], [-27], [-39], [12], [12], [36], [-37], [18], [27], [-40], [-24], [46], [14], [30], [-44], [-22], [-58], [-4], [-61], [-42], [16], [39], [-50], [-48], [-38], [-30], [0], [47], [-5], [-26], [48], [-27], [24], [14], [24], [42], [-59], [62], [-24], [51], [16], [61], [-32], [-40], [22], [-20], [15], [3], [60], [28], [36], [26], [40], [57], [10], [51], [-28], [2], [-36], [-30], [-65], [-23], [-15], [-20], [2], [-12], [-64], [-43], [45], [-27], [-22], [42], [27], [69], [50], [-17], [-30], [-6], [52], [-17], [-20], [4], [-12], [-12], [-34], [16], [-51], [-4], [28], [38], [5], [18], [54], [8], [64], [-29], [5], [19], [0], [5], [2], [54], [46], [69], [-43], [60], [30], [28], [2], [24], [50], [18], [-45], [46], [-54], [36], [-32], [26], [27], [40], [-9], [30], [-50], [70], [15], [-65], [52], [-48], [45], [49], [-45], [10], [19], [14], [-66], [-54], [-38], [0], [44], [46], [70], [-42], [6], [17], [66], [-62], [-22], [15], [80], [46], [-41], [15], [0], [-68], [70], [-78], [-40], [-14], [85], [16], [20], [2], [34], [-55], [32], [45], [-18], [-40], [17], [70], [-65], [-33], [34], [-9], [-43], [-45], [-24], [79], [-4], [32], [16], [-56], [69], [82], [-20], [-84], [-38], [-78], [25], [40], [-26], [2], [18], [48], [44], [42], [0], [-20], [-36], [-54], [-40], [-66], [24], [-54], [-70], [-83], [40], [66], [-14], [-72], [-63], [-70], [-8], [16], [-62], [18], [1], [4], [6], [-42], [12], [90], [66], [0], [40], [30], [-93], [-10], [94], [-55], [-78], [-61], [-45], [39], [-14], [-69], [14], [2], [20], [24], [87], [-10], [4], [13], [-99], [-6], [36], [39], [24], [-37], [2], [72], [-58], [89], [-60], [-29], [28], [-6], [102], [-49], [-93], [24], [88], [-84], [-55], [-81], [-30], [73], [28], [-75], [4], [-75], [-54], [-72], [54], [-91], [-78], [45], [-61], [14], [-19], [18], [-30], [36]];
    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_l();".
// 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_l(: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_l();".
// 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_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1110_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-3, 1]>,<11,R![-1, 1]>,<13,R![-1, 1]>],Snew);
    return Vf;
end function;