LMFDB - Newform orbit 209.2.j.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [209,2,Mod(23,209)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(209, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("209.23");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 209 = 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	209.j (of order $9$, degree $6$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.66887340224$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/209\mathbb{Z}\right)^\times$.

$n$	$78$	$134$
$\chi(n)$	$-\zeta_{18}^{5}$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

23.1

−0.173648	−	0.984808i
−0.173648	+	0.984808i
0.939693	−	0.342020i
0.939693	+	0.342020i
−0.766044	−	0.642788i
−0.766044	+	0.642788i

1.26604

0.460802i

−0.347296

−

1.96962i

−0.141559

−

0.118782i

−0.358441

0.300767i

0.467911

−

2.65366i

1.17365

−

2.03282i

−1.47178

−

2.54920i

−0.939693

0.342020i

−0.592396

0.215615i

100.1

1.26604

−

0.460802i

−0.347296

1.96962i

−0.141559

0.118782i

−0.358441

−

0.300767i

0.467911

2.65366i

1.17365

2.03282i

−1.47178

2.54920i

−0.939693

−

0.342020i

−0.592396

−

0.215615i

111.1

0.673648

0.565258i

1.87939

−

0.684040i

−0.213011

−

1.20805i

−0.286989

1.62760i

1.65270

0.601535i

0.0603074

0.104455i

1.41875

−

2.45734i

0.766044

−

0.642788i

−1.11334

0.934204i

177.1

0.673648

−

0.565258i

1.87939

0.684040i

−0.213011

1.20805i

−0.286989

−

1.62760i

1.65270

−

0.601535i

0.0603074

−

0.104455i

1.41875

2.45734i

0.766044

0.642788i

−1.11334

−

0.934204i

188.1

−0.439693

2.49362i

−1.53209

−

1.28558i

−4.14543

−

1.50881i

3.64543

−

1.32683i

3.87939

−

3.25519i

1.76604

3.05888i

3.05303

−

5.28801i

0.173648

0.984808i

1.70574

9.67372i

199.1

−0.439693

−

2.49362i

−1.53209

1.28558i

−4.14543

1.50881i

3.64543

1.32683i

3.87939

3.25519i

1.76604

−

3.05888i

3.05303

5.28801i

0.173648

−

0.984808i

1.70574

−

9.67372i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	209.2.j.a	✓	6
19.e	even	9	1	inner	209.2.j.a	✓	6
19.e	even	9	1		3971.2.a.e		3
19.f	odd	18	1		3971.2.a.f		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
209.2.j.a	✓	6	1.a	even	1	1	trivial
209.2.j.a	✓	6	19.e	even	9	1	inner
3971.2.a.e		3	19.e	even	9	1
3971.2.a.f		3	19.f	odd	18	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} - 3T_{2}^{5} + 9T_{2}^{4} - 24T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9 $ T2^6 - 3*T2^5 + 9*T2^4 - 24*T2^3 + 36*T2^2 - 27*T2 + 9 acting on $S_{2}^{\mathrm{new}}(209, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} - 3 T^{5} + 9 T^{4} - 24 T^{3} + \cdots + 9 $ T^6 - 3T^5 + 9T^4 - 24T^3 + 36T^2 - 27*T + 9
$3$	$ T^{6} - 8T^{3} + 64 $ T^6 - 8*T^3 + 64
$5$	$ T^{6} - 6 T^{5} + 9 T^{4} - 3 T^{3} + \cdots + 9 $ T^6 - 6T^5 + 9T^4 - 3T^3 + 36T^2 + 27*T + 9
$7$	$ T^{6} - 6 T^{5} + 27 T^{4} - 52 T^{3} + \cdots + 1 $ T^6 - 6T^5 + 27T^4 - 52T^3 + 75T^2 - 9*T + 1
$11$	$ (T^{2} + T + 1)^{3} $ (T^2 + T + 1)^3
$13$	$ T^{6} - 12 T^{5} + 75 T^{4} + \cdots + 5041 $ T^6 - 12T^5 + 75T^4 - 379T^3 + 1572T^2 - 4047*T + 5041
$17$	$ T^{6} + 3 T^{5} - 9 T^{4} + \cdots + 12321 $ T^6 + 3T^5 - 9T^4 - 192T^3 + 522T^2 - 999*T + 12321
$19$	$ T^{6} - 6 T^{5} - 12 T^{4} + \cdots + 6859 $ T^6 - 6T^5 - 12T^4 + 169T^3 - 228T^2 - 2166*T + 6859
$23$	$ T^{6} - 9 T^{5} + 36 T^{4} - 153 T^{3} + \cdots + 81 $ T^6 - 9T^5 + 36T^4 - 153T^3 + 567T^2 - 324*T + 81
$29$	$ T^{6} + 6 T^{5} - 240 T^{3} + \cdots + 576 $ T^6 + 6T^5 - 240T^3 + 576*T^2 + 576
$31$	$ T^{6} + 3 T^{5} + 45 T^{4} + \cdots + 16129 $ T^6 + 3T^5 + 45T^4 + 146T^3 + 1677T^2 + 4572*T + 16129
$37$	$ (T^{3} + 9 T^{2} + 24 T + 19)^{2} $ (T^3 + 9T^2 + 24T + 19)^2
$41$	$ T^{6} + 9 T^{5} + 36 T^{4} + 72 T^{3} + \cdots + 81 $ T^6 + 9T^5 + 36T^4 + 72T^3 + 81T^2 + 81*T + 81
$43$	$ T^{6} + 12 T^{5} + 159 T^{4} + \cdots + 5329 $ T^6 + 12T^5 + 159T^4 + 881T^3 + 2532T^2 - 8103*T + 5329
$47$	$ T^{6} - 27 T^{5} + 324 T^{4} + \cdots + 210681 $ T^6 - 27T^5 + 324T^4 - 2457T^3 + 15309T^2 - 74358*T + 210681
$53$	$ T^{6} - 18 T^{5} + 270 T^{4} + \cdots + 263169 $ T^6 - 18T^5 + 270T^4 - 3375T^3 + 29646T^2 - 138510*T + 263169
$59$	$ T^{6} + 6 T^{5} + 126 T^{4} + \cdots + 45369 $ T^6 + 6T^5 + 126T^4 + 219T^3 - 1746T^2 + 3834*T + 45369
$61$	$ T^{6} + 21 T^{5} + 276 T^{4} + \cdots + 11449 $ T^6 + 21T^5 + 276T^4 + 2114T^3 + 12144T^2 + 20544*T + 11449
$67$	$ T^{6} - 12 T^{5} + 246 T^{4} + \cdots + 128881 $ T^6 - 12T^5 + 246T^4 - 3412T^3 + 24018T^2 - 85083*T + 128881
$71$	$ T^{6} + 42 T^{5} + 693 T^{4} + \cdots + 3249 $ T^6 + 42T^5 + 693T^4 + 5349T^3 + 18396T^2 + 3591*T + 3249
$73$	$ T^{6} + 15 T^{5} + 228 T^{4} + \cdots + 1129969 $ T^6 + 15T^5 + 228T^4 + 539T^3 - 1029T^2 - 216852*T + 1129969
$79$	$ T^{6} + 9 T^{5} + 18 T^{4} - 388 T^{3} + \cdots + 289 $ T^6 + 9T^5 + 18T^4 - 388T^3 + 900T^2 - 306*T + 289
$83$	$ T^{6} - 6 T^{5} + 135 T^{4} + \cdots + 47961 $ T^6 - 6T^5 + 135T^4 + 1032T^3 + 8487T^2 + 21681*T + 47961
$89$	$ T^{6} - 21 T^{5} + 207 T^{4} + \cdots + 12321 $ T^6 - 21T^5 + 207T^4 - 1428T^3 + 5436T^2 + 12987*T + 12321
$97$	$ T^{6} - 27 T^{5} + 378 T^{4} + \cdots + 157609 $ T^6 - 27T^5 + 378T^4 - 3556T^3 + 22626T^2 - 85752*T + 157609
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 209 = 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	209.j (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} + 1) q^{2} + 2 \zeta_{18} q^{3} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18} - 2) q^{4} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{2} + 1) q^{5} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{6} + (\zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{3} - \zeta_{18} + 2) q^{7} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18}) q^{8} + \zeta_{18}^{2} q^{9}+O(q^{10}) \) q + (-z^5 + z^4 - z^3 + z^2 - z + 1) * q^2 + 2z q^3 + (-z^5 + z^4 + z^3 + z - 2) * q^4 + (z^5 - 2z^4 - z^2 + 1) q^5 + (2z^5 - 2z^4 - 2z^2 + 2z + 2) * q^6 + (z^5 + z^4 - 2z^3 - z + 2) q^7 + (3z^5 - 2z^4 + 2z^3 - 2z^2 + 3z) q^8 + z^2 * q^9	\( q + ( - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} + 1) q^{2} + 2 \zeta_{18} q^{3} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18} - 2) q^{4} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{2} + 1) q^{5} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{6} + (\zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{3} - \zeta_{18} + 2) q^{7} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18}) q^{8} + \zeta_{18}^{2} q^{9} + (2 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 4 \zeta_{18} + 2) q^{10} - \zeta_{18}^{3} q^{11} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 4 \zeta_{18} + 2) q^{12} + ( - 2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + \zeta_{18} + 1) q^{13} + ( - 3 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{14} + ( - 4 \zeta_{18}^{5} + 2 \zeta_{18} - 2) q^{15} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{3} - \zeta_{18} + 3) q^{16} + (\zeta_{18}^{5} + 5 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} - 5 \zeta_{18} - 1) q^{17} + ( - \zeta_{18}^{5} + \zeta_{18}^{2} + \zeta_{18} - 1) q^{18} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 4 \zeta_{18} + 2) q^{19} + ( - 7 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 3 \zeta_{18}^{2} + 3 \zeta_{18} - 5) q^{20} + (2 \zeta_{18}^{5} - 4 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 4 \zeta_{18} - 2) q^{21} + ( - \zeta_{18}^{2} + \zeta_{18} - 1) q^{22} + (3 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18} + 1) q^{23} + ( - 4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 6 \zeta_{18}^{2} - 6) q^{24} + (\zeta_{18}^{5} - 5 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} + 1) q^{25} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - 3 \zeta_{18} + 2) q^{26} - 4 \zeta_{18}^{3} q^{27} + ( - 4 \zeta_{18}^{4} + 6 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 6 \zeta_{18} - 4) q^{28} + ( - 2 \zeta_{18}^{3} - 4 \zeta_{18}^{2} - 2 \zeta_{18}) q^{29} + (4 \zeta_{18}^{5} - 8 \zeta_{18}^{4} + 6 \zeta_{18}^{3} - 8 \zeta_{18}^{2} + 4 \zeta_{18}) q^{30} + ( - 4 \zeta_{18}^{5} - 4 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} + 3 \zeta_{18} - 1) q^{31} + (3 \zeta_{18}^{4} - 3 \zeta_{18} - 3) q^{32} - 2 \zeta_{18}^{4} q^{33} + ( - 3 \zeta_{18}^{5} + 5 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - \zeta_{18} - 4) q^{34} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 5 \zeta_{18} + 5) q^{35} + (\zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18} - 1) q^{36} + ( - \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 3) q^{37} + ( - 4 \zeta_{18}^{5} + 7 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 4 \zeta_{18} - 2) q^{38} + ( - 6 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 2 \zeta_{18} + 4) q^{39} + (\zeta_{18}^{5} - 9 \zeta_{18}^{4} + 8 \zeta_{18}^{3} - 8 \zeta_{18}^{2} + 9 \zeta_{18} - 1) q^{40} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} - \zeta_{18}^{2} - 2) q^{41} + (8 \zeta_{18}^{5} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 2 \zeta_{18} + 6) q^{42} + ( - 3 \zeta_{18}^{5} - 6 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{43} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} + 1) q^{44} + ( - 2 \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} + 2) q^{45} + ( - 3 \zeta_{18}^{5} + 6 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 6 \zeta_{18}^{2} - 3 \zeta_{18}) q^{46} + (3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{47} + ( - 6 \zeta_{18}^{4} + 6 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 6 \zeta_{18} - 6) q^{48} + ( - \zeta_{18}^{5} + 4 \zeta_{18}^{4} + \zeta_{18}^{3} + 4 \zeta_{18}^{2} - \zeta_{18}) q^{49} + (\zeta_{18}^{5} + \zeta_{18}^{4} - 7 \zeta_{18}^{3} + 9 \zeta_{18}^{2} - 10 \zeta_{18} + 7) q^{50} + (10 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 10 \zeta_{18}^{2} - 2 \zeta_{18} - 2) q^{51} + ( - 4 \zeta_{18}^{5} + 7 \zeta_{18}^{4} + 4 \zeta_{18}^{2} - 4) q^{52} + (3 \zeta_{18}^{5} + 6 \zeta_{18}^{4} + 6 \zeta_{18}^{3} - 6 \zeta_{18}) q^{53} + ( - 4 \zeta_{18}^{2} + 4 \zeta_{18} - 4) q^{54} + (2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18}) q^{55} + (9 \zeta_{18}^{5} - 8 \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} + 5) q^{56} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 8 \zeta_{18}^{2} + 4 \zeta_{18} - 2) q^{57} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{2} - 4 \zeta_{18}) q^{58} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} - 6 \zeta_{18}^{3} + 6 \zeta_{18}^{2} - \zeta_{18} + 2) q^{59} + (8 \zeta_{18}^{5} - 8 \zeta_{18}^{3} + 6 \zeta_{18}^{2} - 10 \zeta_{18} + 14) q^{60} + (6 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} + 3 \zeta_{18} - 4) q^{61} + (3 \zeta_{18}^{5} - 8 \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{62} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{2} - \zeta_{18} - 1) q^{63} + (4 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3 \zeta_{18} - 4) q^{64} + (3 \zeta_{18}^{5} - 7 \zeta_{18}^{4} + \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 3 \zeta_{18}) q^{65} + ( - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2 \zeta_{18}) q^{66} + ( - \zeta_{18}^{4} + 6 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 6 \zeta_{18} - 1) q^{67} + (3 \zeta_{18}^{5} - 8 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 8 \zeta_{18}^{2} + 3 \zeta_{18}) q^{68} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 6 \zeta_{18}^{3} - 4 \zeta_{18}^{2} + 2 \zeta_{18} - 6) q^{69} + ( - 12 \zeta_{18}^{5} + 13 \zeta_{18}^{4} - 8 \zeta_{18}^{3} + 12 \zeta_{18}^{2} - 5 \zeta_{18} - 5) q^{70} + ( - 7 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 7 \zeta_{18}^{2} - 7) q^{71} + (2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 3 \zeta_{18} + 2) q^{72} + (\zeta_{18}^{5} - \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 11 \zeta_{18} - 2) q^{73} + (3 \zeta_{18}^{5} - 4 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 4 \zeta_{18} - 3) q^{74} + ( - 10 \zeta_{18}^{5} + 8 \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 2 \zeta_{18} - 2) q^{75} + ( - 3 \zeta_{18}^{5} - 6 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 9 \zeta_{18} - 4) q^{76} + ( - \zeta_{18}^{5} + \zeta_{18}^{2} + \zeta_{18} - 2) q^{77} + ( - 4 \zeta_{18}^{5} - 4 \zeta_{18}^{4} + 6 \zeta_{18}^{3} - 6 \zeta_{18}^{2} + 4 \zeta_{18} + 4) q^{78} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{3} + 4 \zeta_{18} - 3) q^{79} + (8 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 7 \zeta_{18} + 10) q^{80} - 11 \zeta_{18}^{4} q^{81} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2}) q^{82} + (4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 3 \zeta_{18} + 2) q^{83} + ( - 8 \zeta_{18}^{5} + 12 \zeta_{18}^{4} - 10 \zeta_{18}^{3} + 12 \zeta_{18}^{2} - 8 \zeta_{18}) q^{84} + (4 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 9 \zeta_{18}^{2} - 2 \zeta_{18} + 4) q^{85} + ( - 4 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18} - 4) q^{86} + ( - 4 \zeta_{18}^{4} - 8 \zeta_{18}^{3} - 4 \zeta_{18}^{2}) q^{87} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 2 \zeta_{18} + 2) q^{88} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - 5 \zeta_{18}^{3} + \zeta_{18}^{2} + 6 \zeta_{18} + 6) q^{89} + ( - 4 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2) q^{90} + (\zeta_{18}^{4} + \zeta_{18}^{3} - 7 \zeta_{18} + 6) q^{91} + ( - 5 \zeta_{18}^{5} + 5 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 6 \zeta_{18} - 10) q^{92} + ( - 8 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 6 \zeta_{18}^{3} + 6 \zeta_{18}^{2} - 2 \zeta_{18} + 8) q^{93} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{2} - 3 \zeta_{18} + 12) q^{94} + (7 \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{2} - 6 \zeta_{18} + 4) q^{95} + (6 \zeta_{18}^{5} - 6 \zeta_{18}^{2} - 6 \zeta_{18}) q^{96} + ( - 3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 2 \zeta_{18} + 3) q^{97} + ( - 5 \zeta_{18}^{5} + 5 \zeta_{18}^{3} + \zeta_{18}^{2} + 6 \zeta_{18} - 4) q^{98} - \zeta_{18}^{5} q^{99} +O(q^{100}) \) q + (-z^5 + z^4 - z^3 + z^2 - z + 1) * q^2 + 2z q^3 + (-z^5 + z^4 + z^3 + z - 2) * q^4 + (z^5 - 2z^4 - z^2 + 1) q^5 + (2z^5 - 2z^4 - 2z^2 + 2z + 2) * q^6 + (z^5 + z^4 - 2z^3 - z + 2) q^7 + (3z^5 - 2z^4 + 2z^3 - 2z^2 + 3z) q^8 + z^2 * q^9 + (2z^4 - 4z^3 + 3z^2 - 4z + 2) * q^10 - z^3 * q^11 + (2z^5 + 2z^4 - 2z^3 + 2z^2 - 4z + 2) q^12 + (-2z^5 - 3z^4 + 2z^3 + 2z^2 + z + 1) * q^13 + (-3z^5 + 4z^4 - 2z^3 + z^2 - 1) q^14 + (-4z^5 + 2z - 2) * q^15 + (3z^5 - 3z^3 - z + 3) * q^16 + (z^5 + 5z^4 + z^3 - z^2 - 5z - 1) * q^17 + (-z^5 + z^2 + z - 1) * q^18 + (z^5 + 2z^4 - 2z^3 - 2z^2 - 4z + 2) * q^19 + (-7z^5 + 4z^4 + 3z^2 + 3z - 5) * q^20 + (2z^5 - 4z^4 + 2z^3 - 2z^2 + 4z - 2) q^21 + (-z^2 + z - 1) * q^22 + (3z^5 + z^4 + z^3 - 2z + 1) * q^23 + (-4z^5 + 4z^4 + 2z^3 + 6z^2 - 6) * q^24 + (z^5 - 5z^4 + 4z^3 - z^2 + z + 1) * q^25 + (-2z^5 - 2z^4 - 2z^3 + 5z^2 - 3z + 2) q^26 - 4z^3 q^27 + (-4z^4 + 6z^3 - 5z^2 + 6z - 4) * q^28 + (-2z^3 - 4z^2 - 2z) q^29 + (4z^5 - 8z^4 + 6z^3 - 8z^2 + 4z) q^30 + (-4z^5 - 4z^4 + z^3 + z^2 + 3z - 1) q^31 + (3z^4 - 3z - 3) * q^32 - 2z^4 q^33 + (-3z^5 + 5z^4 + 5z^3 - z - 4) q^34 + (3z^5 - 3z^3 + 2z^2 - 5z + 5) * q^35 + (z^5 - z^4 + 2z^3 - 2z^2 + z - 1) * q^36 + (-z^4 + z^2 + z - 3) * q^37 + (-4z^5 + 7z^4 - z^3 + z^2 - 4z - 2) q^38 + (-6z^5 + 4z^4 + 2z^2 + 2z + 4) * q^39 + (z^5 - 9z^4 + 8z^3 - 8z^2 + 9z - 1) * q^40 + (-z^5 + z^3 - z^2 - 2) * q^41 + (8z^5 - 4z^4 - 4z^3 - 2z + 6) * q^42 + (-3z^5 - 6z^4 - 2z^3 + z^2 - 1) q^43 + (z^5 - 2z^4 + z^3 - z^2 + z + 1) q^44 + (-2z^3 + z^2 - z + 2) q^45 + (-3z^5 + 6z^4 - 3z^3 + 6z^2 - 3z) q^46 + (3z^4 + 3z^3 - 3z^2 + 3z + 3) * q^47 + (-6z^4 + 6z^3 - 2z^2 + 6z - 6) * q^48 + (-z^5 + 4z^4 + z^3 + 4z^2 - z) * q^49 + (z^5 + z^4 - 7z^3 + 9z^2 - 10z + 7) q^50 + (10z^5 + 2z^4 - 10z^2 - 2z - 2) * q^51 + (-4z^5 + 7z^4 + 4z^2 - 4) q^52 + (3z^5 + 6z^4 + 6z^3 - 6z) * q^53 + (-4z^2 + 4z - 4) * q^54 + (2z^4 - z^3 + z^2 - 2z) * q^55 + (9z^5 - 8z^4 - z^2 - z + 5) * q^56 + (4z^5 - 4z^4 - 2z^3 - 8z^2 + 4z - 2) q^57 + (2z^5 + 2z^4 - 4z^2 - 4z) * q^58 + (-2z^5 + z^4 - 6z^3 + 6z^2 - z + 2) q^59 + (8z^5 - 8z^3 + 6z^2 - 10z + 14) * q^60 + (6z^5 + z^4 + z^3 + 3z - 4) * q^61 + (3z^5 - 8z^4 + z^3 - 2z^2 + 2) q^62 + (-2z^5 + z^4 + 2z^2 - z - 1) * q^63 + (4z^3 + 3z^2 - 3z - 4) q^64 + (3z^5 - 7z^4 + z^3 - 7z^2 + 3z) * q^65 + (-2z^3 + 2z^2 - 2z) q^66 + (-z^4 + 6z^3 - 5z^2 + 6z - 1) q^67 + (3z^5 - 8z^4 + 2z^3 - 8z^2 + 3z) q^68 + (2z^5 + 2z^4 + 6z^3 - 4z^2 + 2z - 6) q^69 + (-12z^5 + 13z^4 - 8z^3 + 12z^2 - 5z - 5) q^70 + (-7z^5 + 2z^4 + 7z^2 - 7) q^71 + (2z^5 + z^4 + z^3 - 3z + 2) * q^72 + (z^5 - z^3 - 3z^2 + 11z - 2) * q^73 + (3z^5 - 4z^4 + 2z^3 - 2z^2 + 4z - 3) q^74 + (-10z^5 + 8z^4 + 2z^2 + 2z - 2) * q^75 + (-3z^5 - 6z^4 + 5z^3 - 5z^2 + 9z - 4) q^76 + (-z^5 + z^2 + z - 2) * q^77 + (-4z^5 - 4z^4 + 6z^3 - 6z^2 + 4z + 4) q^78 + (-3z^5 + 3z^3 + 4z - 3) q^79 + (8z^5 - 3z^4 - 3z^3 - 7z + 10) * q^80 - 11z^4 q^81 + (3z^5 - 3z^4 + 3z^3 - 3z^2) * q^82 + (4z^5 + 4z^4 - 2z^3 - 7z^2 + 3z + 2) q^83 + (-8z^5 + 12z^4 - 10z^3 + 12z^2 - 8z) q^84 + (4z^4 - 2z^3 + 9z^2 - 2z + 4) * q^85 + (-4z^4 - 2z^3 + z^2 - 2z - 4) q^86 + (-4z^4 - 8z^3 - 4z^2) q^87 + (-z^5 - z^4 - 2z^3 + 3z^2 - 2z + 2) q^88 + (-z^5 - z^4 - 5z^3 + z^2 + 6z + 6) * q^89 + (-4z^5 + 3z^4 - 2z^3 + 2z^2 - 2) * q^90 + (z^4 + z^3 - 7z + 6) q^91 + (-5z^5 + 5z^3 - 5z^2 + 6z - 10) * q^92 + (-8z^5 + 2z^4 - 6z^3 + 6z^2 - 2z + 8) q^93 + (3z^5 - 3z^2 - 3z + 12) q^94 + (7z^5 + z^4 + 2z^2 - 6z + 4) q^95 + (6z^5 - 6z^2 - 6z) q^96 + (-3z^5 - 2z^4 + 3z^3 - 3z^2 + 2z + 3) q^97 + (-5z^5 + 5z^3 + z^2 + 6z - 4) q^98 - z^5 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} - 9 q^{4} + 6 q^{5} + 12 q^{6} + 6 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 3 * q^2 - 9 * q^4 + 6 * q^5 + 12 * q^6 + 6 * q^7 + 6 * q^8	\( 6 q + 3 q^{2} - 9 q^{4} + 6 q^{5} + 12 q^{6} + 6 q^{7} + 6 q^{8} - 3 q^{11} + 6 q^{12} + 12 q^{13} - 12 q^{14} - 12 q^{15} + 9 q^{16} - 3 q^{17} - 6 q^{18} + 6 q^{19} - 30 q^{20} - 6 q^{21} - 6 q^{22} + 9 q^{23} - 30 q^{24} + 18 q^{25} + 6 q^{26} - 12 q^{27} - 6 q^{28} - 6 q^{29} + 18 q^{30} - 3 q^{31} - 18 q^{32} - 9 q^{34} + 21 q^{35} - 18 q^{37} - 15 q^{38} + 24 q^{39} + 18 q^{40} - 9 q^{41} + 24 q^{42} - 12 q^{43} + 9 q^{44} + 6 q^{45} - 9 q^{46} + 27 q^{47} - 18 q^{48} + 3 q^{49} + 21 q^{50} - 12 q^{51} - 24 q^{52} + 18 q^{53} - 24 q^{54} - 3 q^{55} + 30 q^{56} - 18 q^{57} - 6 q^{59} + 60 q^{60} - 21 q^{61} + 15 q^{62} - 6 q^{63} - 12 q^{64} + 3 q^{65} - 6 q^{66} + 12 q^{67} + 6 q^{68} - 18 q^{69} - 54 q^{70} - 42 q^{71} + 15 q^{72} - 15 q^{73} - 12 q^{74} - 12 q^{75} - 9 q^{76} - 12 q^{77} + 42 q^{78} - 9 q^{79} + 51 q^{80} + 9 q^{82} + 6 q^{83} - 30 q^{84} + 18 q^{85} - 30 q^{86} - 24 q^{87} + 6 q^{88} + 21 q^{89} - 18 q^{90} + 39 q^{91} - 45 q^{92} + 30 q^{93} + 72 q^{94} + 24 q^{95} + 27 q^{97} - 9 q^{98}+O(q^{100}) \) 6 * q + 3 * q^2 - 9 * q^4 + 6 * q^5 + 12 * q^6 + 6 * q^7 + 6 * q^8 - 3 * q^11 + 6 * q^12 + 12 * q^13 - 12 * q^14 - 12 * q^15 + 9 * q^16 - 3 * q^17 - 6 * q^18 + 6 * q^19 - 30 * q^20 - 6 * q^21 - 6 * q^22 + 9 * q^23 - 30 * q^24 + 18 * q^25 + 6 * q^26 - 12 * q^27 - 6 * q^28 - 6 * q^29 + 18 * q^30 - 3 * q^31 - 18 * q^32 - 9 * q^34 + 21 * q^35 - 18 * q^37 - 15 * q^38 + 24 * q^39 + 18 * q^40 - 9 * q^41 + 24 * q^42 - 12 * q^43 + 9 * q^44 + 6 * q^45 - 9 * q^46 + 27 * q^47 - 18 * q^48 + 3 * q^49 + 21 * q^50 - 12 * q^51 - 24 * q^52 + 18 * q^53 - 24 * q^54 - 3 * q^55 + 30 * q^56 - 18 * q^57 - 6 * q^59 + 60 * q^60 - 21 * q^61 + 15 * q^62 - 6 * q^63 - 12 * q^64 + 3 * q^65 - 6 * q^66 + 12 * q^67 + 6 * q^68 - 18 * q^69 - 54 * q^70 - 42 * q^71 + 15 * q^72 - 15 * q^73 - 12 * q^74 - 12 * q^75 - 9 * q^76 - 12 * q^77 + 42 * q^78 - 9 * q^79 + 51 * q^80 + 9 * q^82 + 6 * q^83 - 30 * q^84 + 18 * q^85 - 30 * q^86 - 24 * q^87 + 6 * q^88 + 21 * q^89 - 18 * q^90 + 39 * q^91 - 45 * q^92 + 30 * q^93 + 72 * q^94 + 24 * q^95 + 27 * q^97 - 9 * q^98

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