
// Make newform 9338.2.a.g in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9338_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_9338_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_9338_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_9338_a();"
function MakeCharacter_9338_a()
    N := 9338;
    order := 1;
    char_gens := [5337, 4467, 3221];
    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_9338_a_Hecke(Kf)
    return MakeCharacter_9338_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], [-2], [0], [-1], [4], [-2], [4], [0], [1], [-1], [-10], [-2], [-6], [-4], [2], [6], [-10], [0], [4], [8], [14], [8], [-12], [-12], [16], [-2], [-8], [12], [-6], [6], [-8], [-6], [-6], [14], [14], [4], [0], [-24], [-14], [-6], [0], [-16], [-8], [-6], [-10], [24], [12], [-6], [-24], [12], [-10], [-12], [-16], [-16], [-30], [8], [-30], [10], [10], [-10], [8], [-12], [-26], [-18], [16], [2], [-28], [-22], [0], [-14], [-26], [32], [28], [-2], [36], [-4], [-18], [-34], [38], [-18], [0], [18], [-16], [16], [-14], [24], [30], [-22], [10], [4], [24], [24], [16], [0], [-32], [16], [-38], [-12], [-16], [34], [-8], [6], [24], [-6], [-12], [-2], [14], [-38], [-24], [14], [-38], [-14], [2], [44], [-8], [-22], [-40], [-38], [-18], [36], [12], [-34], [-48], [-12], [-22], [18], [10], [42], [0], [-36], [-8], [24], [0], [-42], [14], [-4], [-24], [52], [-12], [-46], [38], [-30], [-4], [-12], [2], [20], [-26], [-22], [18], [16], [18], [-8], [8], [-42], [28], [24], [-24], [42], [52], [-24], [52], [34], [-36], [12], [-50], [-44], [-40], [-54], [34], [22], [44], [2], [-48], [-38], [-10], [56], [36], [6], [0], [48], [48], [16], [-50], [54], [24], [0], [-46], [-36], [-26], [-42], [2], [8], [36], [54], [-52], [-24], [10], [50], [68], [-18], [-22], [2], [2], [0], [24], [-6], [-4], [24], [-2], [-30], [22], [-66], [-16], [40], [70], [-8], [-36], [-2], [-48], [30], [70], [-40], [-6], [-24], [30], [50], [60], [-72], [12], [68], [-34], [0], [40], [-10], [40], [-12], [4], [-46], [48], [38], [-44], [14], [6], [62], [66], [-2], [30], [60], [-44], [48], [-38], [-8], [-64], [-56], [-18], [-24], [46], [-72], [2], [-2], [0], [68], [34], [-40], [-22], [32], [-14], [-44], [-36], [-50], [-78], [-12], [-40], [24], [-50], [74], [24], [10], [-10], [24], [-6], [40], [28], [44], [22], [0], [54], [42], [-10], [38], [-42], [64], [54], [-12], [-26], [60], [-16], [22], [30], [70], [-24], [2], [56], [60], [-20], [2], [16], [20], [-32], [-26], [38], [14], [-42], [84], [-72], [16], [-54], [40], [64], [50], [28], [-26], [-2], [40], [-54], [-20], [-20], [26], [24], [74], [-46], [-18], [50], [46], [-52], [50], [58], [-52], [18], [20], [74], [-50], [-66], [-40], [-70], [-16], [-42], [44], [-48], [-12], [-6], [58], [82], [20], [-62], [-82], [12], [40], [-66], [8], [70], [78], [-76], [36], [42], [26], [-82], [8], [22], [-96], [-42], [-84], [74], [52], [62], [-42], [-72], [24], [-84], [-94], [68], [58], [20], [-14], [56], [-14], [-52], [42], [-36], [-60], [10], [-14], [52], [-62], [0], [22], [-68], [16], [100], [18], [62], [78], [-40], [4], [-34], [-98], [-18], [42], [-28], [-50], [-46], [22], [60], [44], [-6], [18], [-24], [-82], [56], [-48], [-6], [36], [-32], [22], [-18], [70], [-68], [26], [30], [-94]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9338_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9338_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_9338_2_a_g(:prec:=1)
    chi := MakeCharacter_9338_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_9338_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_9338_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9338_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![0, 1]>,<11,R![-4, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;