
// Make newform 9338.2.a.f in Magma, downloaded from the LMFDB on 18 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_f();".
// 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_f();".
// 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], [-2], [1], [-6], [-2], [-6], [-8], [-1], [1], [-6], [-8], [-4], [-2], [-2], [2], [0], [-2], [-4], [0], [4], [10], [-4], [2], [-2], [-16], [0], [-12], [10], [0], [20], [-10], [-8], [0], [14], [-16], [-18], [-16], [-12], [-2], [-12], [-6], [6], [14], [-6], [-8], [8], [4], [-12], [-22], [-10], [24], [18], [8], [30], [-2], [0], [-22], [10], [-30], [-20], [-22], [-2], [-18], [22], [-6], [-16], [-12], [12], [26], [-6], [-14], [0], [14], [34], [-24], [24], [-18], [-22], [12], [-20], [28], [-12], [-34], [24], [20], [-14], [-22], [-24], [16], [8], [0], [-32], [-4], [-12], [28], [26], [10], [28], [18], [12], [2], [-12], [4], [0], [12], [-40], [26], [8], [28], [6], [2], [12], [-32], [-12], [4], [28], [20], [-30], [-42], [-38], [14], [-18], [-36], [36], [-26], [6], [-16], [-32], [-22], [36], [-38], [-2], [-12], [-18], [-14], [-50], [28], [-14], [14], [28], [34], [-12], [-6], [20], [-20], [44], [6], [-34], [-16], [-14], [14], [4], [-30], [-18], [22], [-44], [-34], [58], [50], [8], [-46], [4], [-56], [-26], [-36], [-24], [-12], [38], [-4], [0], [-30], [-60], [-16], [-56], [26], [44], [-6], [44], [-14], [-24], [-52], [-26], [6], [4], [-6], [-26], [-40], [10], [0], [-42], [-20], [-10], [-28], [-6], [54], [8], [54], [-18], [0], [-64], [-32], [0], [-34], [-56], [-2], [0], [-36], [10], [62], [-58], [-22], [-22], [-24], [14], [26], [-32], [56], [48], [26], [18], [-48], [54], [-74], [24], [-14], [26], [-36], [48], [36], [-18], [-20], [38], [62], [-52], [-72], [-46], [-60], [-36], [-38], [26], [-40], [40], [30], [-6], [-8], [12], [12], [-72], [-42], [28], [70], [30], [54], [72], [-52], [20], [-16], [-46], [62], [62], [66], [-26], [-26], [76], [48], [16], [-78], [-50], [54], [-78], [-20], [-10], [-26], [40], [-18], [72], [-70], [36], [-52], [-48], [52], [48], [-26], [-34], [24], [68], [-24], [-40], [-26], [-34], [-12], [-6], [72], [10], [10], [-28], [34], [-28], [-30], [30], [18], [-6], [-64], [46], [62], [-44], [46], [-80], [50], [-56], [-64], [-66], [-28], [76], [-50], [-70], [56], [18], [22], [-56], [38], [-90], [52], [10], [74], [-60], [64], [-18], [-78], [-6], [-78], [-76], [46], [58], [12], [30], [-38], [28], [32], [-2], [6], [12], [60], [-18], [0], [-68], [76], [12], [74], [96], [-54], [-30], [80], [-76], [0], [40], [12], [6], [-72], [62], [62], [-36], [64], [-2], [2], [-72], [16], [4], [-40], [-4], [-48], [88], [16], [82], [10], [54], [-50], [-14], [-2], [26], [6], [-8], [-30], [-28], [24], [-68], [94], [-82], [0], [-8], [74], [-40], [26], [-50], [22], [16], [54], [64], [-70], [42], [22], [-2], [6], [-42], [8], [42], [46], [-68], [-12], [-34], [94], [-12], [56], [64], [88], [88], [-32], [62], [96], [-54], [58], [48], [-54], [34], [-6], [24], [-90], [-16], [10]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : 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![2, 1]>,<11,R![6, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;