LMFDB - Newform orbit 2888.1.bb.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2888,1,Mod(115,2888)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2888, base_ring=CyclotomicField(38))

chi = DirichletCharacter(H, H._module([19, 19, 4]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2888.115");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2888 = 2^{3} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2888.bb (of order $38$, degree $18$, 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:	$1.44129975648$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\Q(\zeta_{38})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} + \cdots + 1 $ x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{19}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{19} - \cdots)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q - \zeta_{38}^{3} q^{2} + ( - \zeta_{38}^{17} + 1) q^{3} + \zeta_{38}^{6} q^{4} + ( - \zeta_{38}^{3} - \zeta_{38}) q^{6} - \zeta_{38}^{9} q^{8} + ( - \zeta_{38}^{17} - \zeta_{38}^{15} + 1) q^{9} + ( - \zeta_{38}^{15} + \zeta_{38}^{8}) q^{11} + \cdots + ( - \zeta_{38}^{15} + \cdots + \zeta_{38}^{4}) q^{99} +O(q^{100}) $ q - z^3 * q^2 + (-z^17 + 1) * q^3 + z^6 * q^4 + (-z^3 - z) * q^6 - z^9 * q^8 + (-z^17 - z^15 + 1) * q^9 + (-z^15 + z^8) * q^11 + (z^6 + z^4) * q^12 + z^12 * q^16 + (z^6 - z) * q^17 + (z^18 - z^3 - z) * q^18 + z^16 * q^19 + (z^18 - z^11) * q^22 + (-z^9 - z^7) * q^24 - z^5 * q^25 + (-z^17 - z^15 - z^13 + 1) * q^27 - z^15 * q^32 + (-z^15 - z^13 + z^8 + z^6) * q^33 + (-z^9 + z^4) * q^34 + (z^6 + z^4 + z^2) * q^36 + q^38 + (z^12 + z^10) * q^41 + (-z^7 - z) * q^43 + (z^14 + z^2) * q^44 + (z^12 + z^10) * q^48 - z^7 * q^49 + z^8 * q^50 + (z^18 + z^6 + z^4 - z) * q^51 + (z^18 + z^16 - z^3 - z) * q^54 + (z^16 + z^14) * q^57 + (z^14 + z^8) * q^59 + z^18 * q^64 + (z^18 + z^16 - z^11 - z^9) * q^66 + (-z^11 - z^7) * q^67 + (z^12 - z^7) * q^68 + (-z^9 - z^7 - z^5) * q^72 + (-z^5 + z^2) * q^73 + (-z^5 - z^3) * q^75 - z^3 * q^76 + (-z^17 - z^15 - z^13 - z^11 + 1) * q^81 + (-z^15 - z^13) * q^82 + (z^14 + 1) * q^83 + (z^10 + z^4) * q^86 + (-z^17 - z^5) * q^88 + (z^18 - z^13) * q^89 + (-z^15 - z^13) * q^96 + 2z^10 q^97 + z^10 * q^98 + (-z^15 - z^13 - z^11 + z^8 + z^6 + z^4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - q^{2} + 17 q^{3} - q^{4} - 2 q^{6} - q^{8} + 16 q^{9} - 2 q^{11} - 2 q^{12} - q^{16} - 2 q^{17} - 3 q^{18} - q^{19} - 2 q^{22} - 2 q^{24} - q^{25} + 15 q^{27} - q^{32} - 4 q^{33} - 2 q^{34}+ \cdots - 6 q^{99}+O(q^{100}) $ 18 * q - q^2 + 17 * q^3 - q^4 - 2 * q^6 - q^8 + 16 * q^9 - 2 * q^11 - 2 * q^12 - q^16 - 2 * q^17 - 3 * q^18 - q^19 - 2 * q^22 - 2 * q^24 - q^25 + 15 * q^27 - q^32 - 4 * q^33 - 2 * q^34 - 3 * q^36 + 18 * q^38 - 2 * q^41 - 2 * q^43 - 2 * q^44 - 2 * q^48 - q^49 - q^50 - 4 * q^51 - 4 * q^54 - 2 * q^57 - 2 * q^59 - q^64 - 4 * q^66 - 2 * q^67 - 2 * q^68 - 3 * q^72 - 2 * q^73 - 2 * q^75 - q^76 + 14 * q^81 - 2 * q^82 + 17 * q^83 - 2 * q^86 - 2 * q^88 - 2 * q^89 - 2 * q^96 - 2 * q^97 - q^98 - 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2888\mathbb{Z}\right)^\times$.

$n$	$1445$	$2167$	$2529$
$\chi(n)$	$-1$	$-1$	$\zeta_{38}^{6}$

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} $

115.1

0.401695	+	0.915773i
0.0825793	−	0.996584i
−0.546948	+	0.837166i
0.879474	−	0.475947i
0.879474	+	0.475947i
−0.546948	−	0.837166i
0.0825793	+	0.996584i
0.401695	−	0.915773i
−0.789141	+	0.614213i
0.986361	−	0.164595i
−0.945817	−	0.324699i
0.677282	+	0.735724i
−0.245485	−	0.969400i
−0.245485	+	0.969400i
0.677282	−	0.735724i
−0.945817	+	0.324699i
0.986361	+	0.164595i
−0.789141	−	0.614213i

0.945817

0.324699i

0.322718

−

0.735724i

0.789141

0.614213i

0.544122

−

0.591074i

0.546948

0.837166i

0.240139

0.260861i

267.1

0.245485

−

0.969400i

0.0136387

0.164595i

−0.879474

−

0.475947i

0.162906

0.0271842i

−0.677282

0.735724i

0.959456

−

0.160105i

419.1

−0.986361

−

0.164595i

0.598305

0.915773i

0.945817

0.324699i

−0.439413

−

1.00176i

−0.879474

−

0.475947i

−0.0789770

0.180049i

571.1

−0.0825793

0.996584i

1.54695

0.837166i

−0.986361

−

0.164595i

−0.962053

1.47253i

0.245485

−

0.969400i

1.14525

1.75294i

875.1

−0.0825793

−

0.996584i

1.54695

−

0.837166i

−0.986361

0.164595i

−0.962053

−

1.47253i

0.245485

0.969400i

1.14525

−

1.75294i

1027.1

−0.986361

0.164595i

0.598305

−

0.915773i

0.945817

−

0.324699i

−0.439413

1.00176i

−0.879474

0.475947i

−0.0789770

−

0.180049i

1179.1

0.245485

0.969400i

0.0136387

−

0.164595i

−0.879474

0.475947i

0.162906

−

0.0271842i

−0.677282

−

0.735724i

0.959456

0.160105i

1331.1

0.945817

−

0.324699i

0.322718

0.735724i

0.789141

−

0.614213i

0.544122

0.591074i

0.546948

−

0.837166i

0.240139

−

0.260861i

1483.1

−0.401695

−

0.915773i

1.24549

0.969400i

−0.677282

0.735724i

0.387445

−

1.52999i

0.945817

0.324699i

0.366012

1.44535i

1635.1

−0.879474

0.475947i

1.94582

0.324699i

0.546948

−

0.837166i

−1.86584

0.640542i

−0.0825793

0.996584i

2.73496

0.938912i

1787.1

0.546948

0.837166i

1.78914

−

0.614213i

−0.401695

0.915773i

1.49277

1.16187i

−0.986361

0.164595i

2.03463

−

1.58361i

1939.1

0.789141

−

0.614213i

0.917421

−

0.996584i

0.245485

−

0.969400i

0.111859

−

1.34994i

−0.401695

−

0.915773i

−0.0689406

−

0.831990i

2091.1

−0.677282

−

0.735724i

0.120526

−

0.475947i

−0.0825793

0.996584i

−0.431796

0.233676i

0.789141

−

0.614213i

0.667474

0.361219i

2243.1

−0.677282

0.735724i

0.120526

0.475947i

−0.0825793

−

0.996584i

−0.431796

−

0.233676i

0.789141

0.614213i

0.667474

−

0.361219i

2395.1

0.789141

0.614213i

0.917421

0.996584i

0.245485

0.969400i

0.111859

1.34994i

−0.401695

0.915773i

−0.0689406

0.831990i

2547.1

0.546948

−

0.837166i

1.78914

0.614213i

−0.401695

−

0.915773i

1.49277

−

1.16187i

−0.986361

−

0.164595i

2.03463

1.58361i

2699.1

−0.879474

−

0.475947i

1.94582

−

0.324699i

0.546948

0.837166i

−1.86584

−

0.640542i

−0.0825793

−

0.996584i

2.73496

−

0.938912i

2851.1

−0.401695

0.915773i

1.24549

−

0.969400i

−0.677282

−

0.735724i

0.387445

1.52999i

0.945817

−

0.324699i

0.366012

−

1.44535i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
361.g	even	19	1	inner
2888.bb	odd	38	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2888.1.bb.a	✓	18
8.d	odd	2	1	CM	2888.1.bb.a	✓	18
361.g	even	19	1	inner	2888.1.bb.a	✓	18
2888.bb	odd	38	1	inner	2888.1.bb.a	✓	18

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2888.1.bb.a	✓	18	1.a	even	1	1	trivial
2888.1.bb.a	✓	18	8.d	odd	2	1	CM
2888.1.bb.a	✓	18	361.g	even	19	1	inner
2888.1.bb.a	✓	18	2888.bb	odd	38	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2888, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{18} + T^{17} + \cdots + 1 $ T^18 + T^17 + T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	$ T^{18} - 17 T^{17} + \cdots + 1 $ T^18 - 17T^17 + 137T^16 - 695T^15 + 2486T^14 - 6656T^13 + 13820T^12 - 22748T^11 + 30086T^10 - 32206T^9 + 27966T^8 - 19650T^7 + 11088T^6 - 4956T^5 + 1716T^4 - 444T^3 + 81T^2 - 9*T + 1
$5$	$ T^{18} $ T^18
$7$	$ T^{18} $ T^18
$11$	$ T^{18} + 2 T^{17} + \cdots + 1 $ T^18 + 2T^17 + 4T^16 - 11T^15 - 22T^14 - 44T^13 + 45T^12 + 90T^11 + 180T^10 - 58T^9 - 116T^8 - 251T^7 + 68T^6 + 136T^5 + 139T^4 + 12T^3 + 24T^2 - 9*T + 1
$13$	$ T^{18} $ T^18
$17$	$ T^{18} + 2 T^{17} + \cdots + 1 $ T^18 + 2T^17 + 4T^16 + 8T^15 - 3T^14 - 6T^13 - 12T^12 - 24T^11 + 66T^10 + 56T^9 + 112T^8 + 224T^7 + 220T^6 + 3T^5 - 13T^4 - 26T^3 + 43T^2 - 9*T + 1
$19$	$ T^{18} + T^{17} + \cdots + 1 $ T^18 + T^17 + T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$23$	$ T^{18} $ T^18
$29$	$ T^{18} $ T^18
$31$	$ T^{18} $ T^18
$37$	$ T^{18} $ T^18
$41$	$ T^{18} + 2 T^{17} + \cdots + 1 $ T^18 + 2T^17 + 4T^16 + 8T^15 + 16T^14 + 32T^13 + 64T^12 + 33T^11 + 66T^10 + 132T^9 + 36T^8 - 23T^7 - 46T^6 - 206T^5 + 25T^4 + 31T^3 + 43T^2 + 10*T + 1
$43$	$ T^{18} + 2 T^{17} + \cdots + 1 $ T^18 + 2T^17 + 4T^16 + 8T^15 + 16T^14 + 32T^13 + 64T^12 + 71T^11 + 9T^10 - T^9 - 2T^8 - 4T^7 - 8T^6 - 35T^5 + 196T^4 - 178T^3 + 62T^2 - 9T + 1
$47$	$ T^{18} $ T^18
$53$	$ T^{18} $ T^18
$59$	$ T^{18} + 2 T^{17} + \cdots + 1 $ T^18 + 2T^17 + 4T^16 + 8T^15 - 3T^14 - 6T^13 - 12T^12 - 24T^11 + 66T^10 + 56T^9 + 112T^8 + 224T^7 + 220T^6 + 3T^5 - 13T^4 - 26T^3 + 43T^2 - 9*T + 1
$61$	$ T^{18} $ T^18
$67$	$ T^{18} + 2 T^{17} + \cdots + 1 $ T^18 + 2T^17 + 4T^16 + 8T^15 + 16T^14 + 32T^13 + 64T^12 + 71T^11 + 9T^10 - T^9 - 2T^8 - 4T^7 - 8T^6 - 35T^5 + 196T^4 - 178T^3 + 62T^2 - 9T + 1
$71$	$ T^{18} $ T^18
$73$	$ T^{18} + 2 T^{17} + \cdots + 1 $ T^18 + 2T^17 + 4T^16 + 8T^15 + 16T^14 - 6T^13 - 12T^12 - 24T^11 - 67T^10 - 134T^9 + 55T^8 + 110T^7 + 201T^6 + 79T^5 + 158T^4 - 7T^3 + 24T^2 + 10*T + 1
$79$	$ T^{18} $ T^18
$83$	$ T^{18} - 17 T^{17} + \cdots + 1 $ T^18 - 17T^17 + 137T^16 - 695T^15 + 2486T^14 - 6656T^13 + 13820T^12 - 22748T^11 + 30086T^10 - 32206T^9 + 27966T^8 - 19650T^7 + 11088T^6 - 4956T^5 + 1716T^4 - 444T^3 + 81T^2 - 9*T + 1
$89$	$ T^{18} + 2 T^{17} + \cdots + 1 $ T^18 + 2T^17 + 4T^16 + 8T^15 - 3T^14 - 6T^13 - 12T^12 - 24T^11 + 66T^10 + 56T^9 + 112T^8 + 224T^7 + 220T^6 + 3T^5 - 13T^4 - 26T^3 + 43T^2 - 9*T + 1
$97$	$ T^{18} + 2 T^{17} + \cdots + 262144 $ T^18 + 2T^17 + 4T^16 + 8T^15 + 16T^14 + 32T^13 + 64T^12 + 128T^11 + 256T^10 + 512T^9 + 1024T^8 + 2048T^7 + 4096T^6 + 8192T^5 + 16384T^4 + 32768T^3 + 65536T^2 + 131072*T + 262144
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2888 = 2^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2888.bb (of order \(38\), degree \(18\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{38}^{3} q^{2} + ( - \zeta_{38}^{17} + 1) q^{3} + \zeta_{38}^{6} q^{4} + ( - \zeta_{38}^{3} - \zeta_{38}) q^{6} - \zeta_{38}^{9} q^{8} + ( - \zeta_{38}^{17} - \zeta_{38}^{15} + 1) q^{9} + ( - \zeta_{38}^{15} + \zeta_{38}^{8}) q^{11} + \cdots + ( - \zeta_{38}^{15} + \cdots + \zeta_{38}^{4}) q^{99} +O(q^{100}) \) q - z^3 * q^2 + (-z^17 + 1) * q^3 + z^6 * q^4 + (-z^3 - z) * q^6 - z^9 * q^8 + (-z^17 - z^15 + 1) * q^9 + (-z^15 + z^8) * q^11 + (z^6 + z^4) * q^12 + z^12 * q^16 + (z^6 - z) * q^17 + (z^18 - z^3 - z) * q^18 + z^16 * q^19 + (z^18 - z^11) * q^22 + (-z^9 - z^7) * q^24 - z^5 * q^25 + (-z^17 - z^15 - z^13 + 1) * q^27 - z^15 * q^32 + (-z^15 - z^13 + z^8 + z^6) * q^33 + (-z^9 + z^4) * q^34 + (z^6 + z^4 + z^2) * q^36 + q^38 + (z^12 + z^10) * q^41 + (-z^7 - z) * q^43 + (z^14 + z^2) * q^44 + (z^12 + z^10) * q^48 - z^7 * q^49 + z^8 * q^50 + (z^18 + z^6 + z^4 - z) * q^51 + (z^18 + z^16 - z^3 - z) * q^54 + (z^16 + z^14) * q^57 + (z^14 + z^8) * q^59 + z^18 * q^64 + (z^18 + z^16 - z^11 - z^9) * q^66 + (-z^11 - z^7) * q^67 + (z^12 - z^7) * q^68 + (-z^9 - z^7 - z^5) * q^72 + (-z^5 + z^2) * q^73 + (-z^5 - z^3) * q^75 - z^3 * q^76 + (-z^17 - z^15 - z^13 - z^11 + 1) * q^81 + (-z^15 - z^13) * q^82 + (z^14 + 1) * q^83 + (z^10 + z^4) * q^86 + (-z^17 - z^5) * q^88 + (z^18 - z^13) * q^89 + (-z^15 - z^13) * q^96 + 2z^10 q^97 + z^10 * q^98 + (-z^15 - z^13 - z^11 + z^8 + z^6 + z^4) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} + 17 q^{3} - q^{4} - 2 q^{6} - q^{8} + 16 q^{9} - 2 q^{11} - 2 q^{12} - q^{16} - 2 q^{17} - 3 q^{18} - q^{19} - 2 q^{22} - 2 q^{24} - q^{25} + 15 q^{27} - q^{32} - 4 q^{33} - 2 q^{34}+ \cdots - 6 q^{99}+O(q^{100}) \) 18 * q - q^2 + 17 * q^3 - q^4 - 2 * q^6 - q^8 + 16 * q^9 - 2 * q^11 - 2 * q^12 - q^16 - 2 * q^17 - 3 * q^18 - q^19 - 2 * q^22 - 2 * q^24 - q^25 + 15 * q^27 - q^32 - 4 * q^33 - 2 * q^34 - 3 * q^36 + 18 * q^38 - 2 * q^41 - 2 * q^43 - 2 * q^44 - 2 * q^48 - q^49 - q^50 - 4 * q^51 - 4 * q^54 - 2 * q^57 - 2 * q^59 - q^64 - 4 * q^66 - 2 * q^67 - 2 * q^68 - 3 * q^72 - 2 * q^73 - 2 * q^75 - q^76 + 14 * q^81 - 2 * q^82 + 17 * q^83 - 2 * q^86 - 2 * q^88 - 2 * q^89 - 2 * q^96 - 2 * q^97 - q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2888.1.bb.a

Level	$2888$
Weight	$1$
Character orbit	2888.bb
Analytic conductor	$1.441$
Analytic rank	$0$
Dimension	$18$
Projective image	$D_{19}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.44129975648\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Coefficient field:	\(\Q(\zeta_{38})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} + \cdots + 1 \) x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{19}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{19} - \cdots)\)

\(n\)	\(1445\)	\(2167\)	\(2529\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\zeta_{38}^{6}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 115.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
361.g	even	19	1	inner
2888.bb	odd	38	1	inner

$p$	$F_p(T)$
$2$	\( T^{18} + T^{17} + \cdots + 1 \) T^18 + T^17 + T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	\( T^{18} - 17 T^{17} + \cdots + 1 \) T^18 - 17T^17 + 137T^16 - 695T^15 + 2486T^14 - 6656T^13 + 13820T^12 - 22748T^11 + 30086T^10 - 32206T^9 + 27966T^8 - 19650T^7 + 11088T^6 - 4956T^5 + 1716T^4 - 444T^3 + 81T^2 - 9*T + 1
$5$	\( T^{18} \) T^18
$7$	\( T^{18} \) T^18
$11$	\( T^{18} + 2 T^{17} + \cdots + 1 \) T^18 + 2T^17 + 4T^16 - 11T^15 - 22T^14 - 44T^13 + 45T^12 + 90T^11 + 180T^10 - 58T^9 - 116T^8 - 251T^7 + 68T^6 + 136T^5 + 139T^4 + 12T^3 + 24T^2 - 9*T + 1
$13$	\( T^{18} \) T^18
$17$	\( T^{18} + 2 T^{17} + \cdots + 1 \) T^18 + 2T^17 + 4T^16 + 8T^15 - 3T^14 - 6T^13 - 12T^12 - 24T^11 + 66T^10 + 56T^9 + 112T^8 + 224T^7 + 220T^6 + 3T^5 - 13T^4 - 26T^3 + 43T^2 - 9*T + 1
$19$	\( T^{18} + T^{17} + \cdots + 1 \) T^18 + T^17 + T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$23$	\( T^{18} \) T^18
$29$	\( T^{18} \) T^18
$31$	\( T^{18} \) T^18
$37$	\( T^{18} \) T^18
$41$	\( T^{18} + 2 T^{17} + \cdots + 1 \) T^18 + 2T^17 + 4T^16 + 8T^15 + 16T^14 + 32T^13 + 64T^12 + 33T^11 + 66T^10 + 132T^9 + 36T^8 - 23T^7 - 46T^6 - 206T^5 + 25T^4 + 31T^3 + 43T^2 + 10*T + 1
$43$	\( T^{18} + 2 T^{17} + \cdots + 1 \) T^18 + 2T^17 + 4T^16 + 8T^15 + 16T^14 + 32T^13 + 64T^12 + 71T^11 + 9T^10 - T^9 - 2T^8 - 4T^7 - 8T^6 - 35T^5 + 196T^4 - 178T^3 + 62T^2 - 9T + 1
$47$	\( T^{18} \) T^18
$53$	\( T^{18} \) T^18
$59$	\( T^{18} + 2 T^{17} + \cdots + 1 \) T^18 + 2T^17 + 4T^16 + 8T^15 - 3T^14 - 6T^13 - 12T^12 - 24T^11 + 66T^10 + 56T^9 + 112T^8 + 224T^7 + 220T^6 + 3T^5 - 13T^4 - 26T^3 + 43T^2 - 9*T + 1
$61$	\( T^{18} \) T^18
$67$	\( T^{18} + 2 T^{17} + \cdots + 1 \) T^18 + 2T^17 + 4T^16 + 8T^15 + 16T^14 + 32T^13 + 64T^12 + 71T^11 + 9T^10 - T^9 - 2T^8 - 4T^7 - 8T^6 - 35T^5 + 196T^4 - 178T^3 + 62T^2 - 9T + 1
$71$	\( T^{18} \) T^18
$73$	\( T^{18} + 2 T^{17} + \cdots + 1 \) T^18 + 2T^17 + 4T^16 + 8T^15 + 16T^14 - 6T^13 - 12T^12 - 24T^11 - 67T^10 - 134T^9 + 55T^8 + 110T^7 + 201T^6 + 79T^5 + 158T^4 - 7T^3 + 24T^2 + 10*T + 1
$79$	\( T^{18} \) T^18
$83$	\( T^{18} - 17 T^{17} + \cdots + 1 \) T^18 - 17T^17 + 137T^16 - 695T^15 + 2486T^14 - 6656T^13 + 13820T^12 - 22748T^11 + 30086T^10 - 32206T^9 + 27966T^8 - 19650T^7 + 11088T^6 - 4956T^5 + 1716T^4 - 444T^3 + 81T^2 - 9*T + 1
$89$	\( T^{18} + 2 T^{17} + \cdots + 1 \) T^18 + 2T^17 + 4T^16 + 8T^15 - 3T^14 - 6T^13 - 12T^12 - 24T^11 + 66T^10 + 56T^9 + 112T^8 + 224T^7 + 220T^6 + 3T^5 - 13T^4 - 26T^3 + 43T^2 - 9*T + 1
$97$	\( T^{18} + 2 T^{17} + \cdots + 262144 \) T^18 + 2T^17 + 4T^16 + 8T^15 + 16T^14 + 32T^13 + 64T^12 + 128T^11 + 256T^10 + 512T^9 + 1024T^8 + 2048T^7 + 4096T^6 + 8192T^5 + 16384T^4 + 32768T^3 + 65536T^2 + 131072*T + 262144
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