
// Make newform 1122.2.a.n in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1122_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_1122_2_a_n();".
// 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_1122_2_a_n();".
// 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_1122_a();"
function MakeCharacter_1122_a()
    N := 1122;
    order := 1;
    char_gens := [749, 409, 1057];
    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_1122_a_Hecke(Kf)
    return MakeCharacter_1122_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], [4], [-2], [1], [0], [-1], [0], [-6], [-2], [4], [2], [-6], [-4], [-6], [8], [8], [-8], [-4], [-6], [10], [-6], [-4], [-14], [14], [18], [-12], [4], [-12], [18], [-2], [-4], [6], [-8], [6], [6], [10], [4], [0], [-2], [-24], [-18], [18], [22], [-18], [16], [-12], [-24], [12], [2], [-22], [-4], [-26], [-16], [18], [8], [-12], [-14], [-16], [6], [-4], [-18], [8], [18], [2], [8], [28], [34], [12], [8], [34], [36], [-12], [0], [-20], [30], [-8], [-2], [14], [-10], [-16], [22], [20], [10], [-14], [16], [-18], [-34], [38], [36], [-36], [-12], [0], [36], [-28], [-16], [-8], [30], [36], [-12], [8], [30], [12], [38], [-24], [14], [0], [46], [14], [42], [-22], [-20], [-6], [28], [8], [-26], [12], [-18], [24], [44], [-38], [-10], [38], [-16], [28], [22], [14], [-30], [20], [0], [-4], [24], [-36], [-42], [6], [54], [0], [36], [-48], [-30], [20], [22], [20], [-12], [-2], [-34], [0], [-30], [36], [-6], [-52], [26], [52], [-40], [-4], [-22], [-46], [38], [18], [-42], [-52], [42], [6], [24], [-2], [-54], [44], [4], [-42], [-16], [36], [10], [32], [34], [-8], [50], [-48], [-24], [-46], [-64], [8], [-36], [-22], [-22], [-26], [12], [-32], [-20], [6], [52], [14], [-12], [36], [24], [-44], [10], [-14], [-18], [2], [24], [-6], [22], [-6], [-38], [-64], [-20], [-44], [52], [-26], [4], [-42], [52], [-12], [60], [-48], [-26], [-26], [14], [18], [-12], [-52], [18], [6], [36], [60], [-24], [26], [6], [70], [68], [-26], [4], [-70], [-2], [12], [-24], [10], [50], [-48], [2], [60], [44], [-32], [-30], [-74], [48], [-16], [60], [56], [-44], [-12], [2], [-22], [-50], [54], [-44], [-38], [52], [48], [38], [-2], [-52], [28], [-52], [-42], [76], [44], [-30], [-8], [26], [-66], [52], [46], [6], [26], [-36], [-76], [-44], [-38], [36], [72], [-80], [24], [0], [-8], [-74], [-34], [6], [-44], [-34], [-44], [56], [54], [52], [-12], [10], [12], [32], [-36], [-76], [-74], [6], [-2], [-4], [28], [2], [-60], [-22], [-62], [16], [64], [-40], [-78], [-68], [-24], [34], [-44], [12], [82], [-30], [-40], [6], [-74], [60], [-42], [14], [-12], [28], [-56], [-10], [48], [-60], [-10], [52], [-20], [36], [2], [-74], [-38], [-58], [-62], [38], [-54], [64], [-32], [44], [-10], [-20], [48], [36], [84], [10], [8], [26], [-12], [58], [-58], [36], [90], [-22], [32], [-82], [-42], [92], [28], [-6], [10], [-26], [-66], [16], [-44], [-20], [-6], [70], [-38], [60], [-8], [82], [18], [38], [84], [-18], [-38], [62], [-84], [86], [16], [70], [-44], [66], [-82], [64], [-88], [-4], [10], [6], [-94], [-74], [92], [90], [40], [78], [10], [-10], [-74], [54], [-42], [-54], [4], [-92], [26], [-98], [-84], [-72], [2], [16], [0], [48], [-18], [76], [20], [-104], [-38], [44], [-10], [-72], [-12], [-54], [-76], [60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1122_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1122_2_a_n();".
// 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_1122_2_a_n(:prec:=1)
    chi := MakeCharacter_1122_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_1122_2_a_n();".
// 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_1122_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1122_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, 1]>,<7,R![2, 1]>,<13,R![0, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;