LMFDB - Newform orbit 1096.1.u.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1096,1,Mod(99,1096)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1096, base_ring=CyclotomicField(34))

chi = DirichletCharacter(H, H._module([17, 17, 31]))

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

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

chi := DirichletCharacter("1096.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1096 = 2^{3} \cdot 137 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1096.u (of order $34$, degree $16$, 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:	$0.546975253846$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{34})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 $ 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:	$D_{34}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{34} - \cdots)$

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1096\mathbb{Z}\right)^\times$.

$n$	$549$	$823$	$825$
$\chi(n)$	$-1$	$-1$	$\zeta_{34}^{7}$

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} $

99.1

−0.932472	−	0.361242i
−0.932472	+	0.361242i
0.850217	+	0.526432i
−0.445738	−	0.895163i
0.602635	+	0.798017i
0.982973	−	0.183750i
0.273663	+	0.961826i
0.273663	−	0.961826i
−0.739009	−	0.673696i
−0.739009	+	0.673696i
−0.0922684	+	0.995734i
0.850217	−	0.526432i
0.602635	−	0.798017i
−0.445738	+	0.895163i
0.982973	+	0.183750i
−0.0922684	−	0.995734i

−0.932472

−

0.361242i

1.72198

−

0.489946i

0.739009

0.673696i

−1.78269

−

0.165190i

−0.445738

−

0.895163i

1.87496

−

1.16092i

155.1

−0.932472

0.361242i

1.72198

0.489946i

0.739009

−

0.673696i

−1.78269

0.165190i

−0.445738

0.895163i

1.87496

1.16092i

323.1

0.850217

0.526432i

0.719401

−

1.85699i

0.445738

0.895163i

1.58923

−

1.20013i

−0.0922684

0.995734i

−2.19186

−

1.99815i

339.1

−0.445738

−

0.895163i

0.247582

−

0.271585i

−0.602635

0.798017i

−0.353470

−

0.100571i

0.982973

0.183750i

0.0798070

0.861255i

355.1

0.602635

0.798017i

−0.719401

−

0.0666624i

−0.273663

0.961826i

−0.380338

−

0.614268i

−0.932472

0.361242i

−0.469879

−

0.0878355i

395.1

0.982973

−

0.183750i

0.840204

−

0.634493i

0.932472

−

0.361242i

0.709310

−

0.778076i

0.850217

−

0.526432i

0.0296988

−

0.104380i

475.1

0.273663

0.961826i

−0.247582

−

1.32445i

−0.850217

0.526432i

1.20614

−

0.600584i

−0.739009

−

0.673696i

−0.760397

0.294579i

563.1

0.273663

−

0.961826i

−0.247582

1.32445i

−0.850217

−

0.526432i

1.20614

0.600584i

−0.739009

0.673696i

−0.760397

−

0.294579i

611.1

−0.739009

−

0.673696i

−0.840204

−

1.35698i

0.0922684

0.995734i

−0.293271

1.56886i

0.602635

−

0.798017i

−0.689703

1.38511i

635.1

−0.739009

0.673696i

−0.840204

1.35698i

0.0922684

−

0.995734i

−0.293271

−

1.56886i

0.602635

0.798017i

−0.689703

−

1.38511i

651.1

−0.0922684

0.995734i

−1.72198

0.857445i

−0.982973

−

0.183750i

−0.694903

−

1.79375i

0.273663

−

0.961826i

1.62738

−

2.15499i

699.1

0.850217

−

0.526432i

0.719401

1.85699i

0.445738

−

0.895163i

1.58923

1.20013i

−0.0922684

−

0.995734i

−2.19186

1.99815i

707.1

0.602635

−

0.798017i

−0.719401

0.0666624i

−0.273663

−

0.961826i

−0.380338

0.614268i

−0.932472

−

0.361242i

−0.469879

0.0878355i

763.1

−0.445738

0.895163i

0.247582

0.271585i

−0.602635

−

0.798017i

−0.353470

0.100571i

0.982973

−

0.183750i

0.0798070

−

0.861255i

899.1

0.982973

0.183750i

0.840204

0.634493i

0.932472

0.361242i

0.709310

0.778076i

0.850217

0.526432i

0.0296988

0.104380i

963.1

−0.0922684

−

0.995734i

−1.72198

−

0.857445i

−0.982973

0.183750i

−0.694903

1.79375i

0.273663

0.961826i

1.62738

2.15499i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
137.f	even	34	1	inner
1096.u	odd	34	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1096.1.u.a	✓	16
8.d	odd	2	1	CM	1096.1.u.a	✓	16
137.f	even	34	1	inner	1096.1.u.a	✓	16
1096.u	odd	34	1	inner	1096.1.u.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1096.1.u.a	✓	16	1.a	even	1	1	trivial
1096.1.u.a	✓	16	8.d	odd	2	1	CM
1096.1.u.a	✓	16	137.f	even	34	1	inner
1096.1.u.a	✓	16	1096.u	odd	34	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1096, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} - T^{15} + \cdots + 1 $ 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^{16} - 17 T^{11} + \cdots + 17 $ T^16 - 17T^11 + 119T^8 + 68T^6 - 85T^5 + 221T^3 + 17T^2 - 34*T + 17
$5$	$ T^{16} $ T^16
$7$	$ T^{16} $ T^16
$11$	$ T^{16} - 2 T^{15} + \cdots + 1 $ T^16 - 2T^15 + 4T^14 - 8T^13 - T^12 + 2T^11 - 4T^10 + 25T^9 + 35T^8 - 70T^7 + 140T^6 - 59T^5 - T^4 + 2T^3 + 30T^2 + 8*T + 1
$13$	$ T^{16} $ T^16
$17$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 47T^10 + 9T^9 + T^8 + 2T^7 + 4T^6 + 8T^5 + 67T^4 - 104T^3 + 47T^2 - 8T + 1
$19$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 64T^10 + 128T^9 + 256T^8 + 495T^7 + 786T^6 + 858T^5 + 594T^4 + 253T^3 + 64T^2 + 9*T + 1
$23$	$ T^{16} $ T^16
$29$	$ T^{16} $ T^16
$31$	$ T^{16} $ T^16
$37$	$ T^{16} $ T^16
$41$	$ T^{16} + 17 T^{14} + \cdots + 17 $ T^16 + 17T^14 + 119T^12 + 442T^10 + 935T^8 + 1122T^6 + 714T^4 + 204*T^2 + 17
$43$	$ T^{16} + 51 T^{9} + \cdots + 17 $ T^16 + 51T^9 - 238T^7 + 255T^5 + 17T^4 - 102T^3 + 85T^2 + 17*T + 17
$47$	$ T^{16} $ T^16
$53$	$ T^{16} $ T^16
$59$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 - 2T^11 - 4T^10 - 8T^9 - 16T^8 - 100T^7 + 21T^6 + 25T^5 + 50T^4 + 83T^3 + 47T^2 + 9*T + 1
$61$	$ T^{16} $ T^16
$67$	$ T^{16} + 51 T^{9} + \cdots + 17 $ T^16 + 51T^9 - 238T^7 + 255T^5 + 17T^4 - 102T^3 + 85T^2 + 17*T + 17
$71$	$ T^{16} $ T^16
$73$	$ T^{16} - 2 T^{15} + \cdots + 1 $ T^16 - 2T^15 + 4T^14 - 8T^13 - T^12 + 2T^11 - 4T^10 + 25T^9 + 35T^8 - 70T^7 + 140T^6 - 59T^5 - T^4 + 2T^3 + 30T^2 + 8*T + 1
$79$	$ T^{16} $ T^16
$83$	$ T^{16} + 17 T^{12} + \cdots + 17 $ T^16 + 17T^12 + 17T^9 + 85T^8 - 221T^5 + 119T^4 + 34T^2 + 68*T + 17
$89$	$ T^{16} + 51 T^{9} + \cdots + 17 $ T^16 + 51T^9 - 238T^7 + 255T^5 + 17T^4 - 102T^3 + 85T^2 + 17*T + 17
$97$	$ T^{16} - 17 T^{15} + \cdots + 17 $ T^16 - 17T^15 + 136T^14 - 680T^13 + 2380T^12 - 6188T^11 + 12376T^10 - 19448T^9 + 24310T^8 - 24310T^7 + 19448T^6 - 12376T^5 + 6188T^4 - 2380T^3 + 680T^2 - 136*T + 17
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Classical	Maass
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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:	\( N \)	\(=\)	\( 1096 = 2^{3} \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1096.u (of order \(34\), degree \(16\), minimal)

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

Introduction

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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	1096.1.u.a

Level	$1096$
Weight	$1$
Character orbit	1096.u
Analytic conductor	$0.547$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{34}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.546975253846\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\Q(\zeta_{34})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 \) 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:	\(D_{34}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{34} - \cdots)\)

\(n\)	\(549\)	\(823\)	\(825\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\zeta_{34}^{7}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 99.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
137.f	even	34	1	inner
1096.u	odd	34	1	inner

$p$	$F_p(T)$
$2$	\( T^{16} - T^{15} + \cdots + 1 \) 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
$3$	\( T^{16} - 17 T^{11} + \cdots + 17 \) T^16 - 17T^11 + 119T^8 + 68T^6 - 85T^5 + 221T^3 + 17T^2 - 34*T + 17
$5$	\( T^{16} \) T^16
$7$	\( T^{16} \) T^16
$11$	\( T^{16} - 2 T^{15} + \cdots + 1 \) T^16 - 2T^15 + 4T^14 - 8T^13 - T^12 + 2T^11 - 4T^10 + 25T^9 + 35T^8 - 70T^7 + 140T^6 - 59T^5 - T^4 + 2T^3 + 30T^2 + 8*T + 1
$13$	\( T^{16} \) T^16
$17$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 47T^10 + 9T^9 + T^8 + 2T^7 + 4T^6 + 8T^5 + 67T^4 - 104T^3 + 47T^2 - 8T + 1
$19$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 64T^10 + 128T^9 + 256T^8 + 495T^7 + 786T^6 + 858T^5 + 594T^4 + 253T^3 + 64T^2 + 9*T + 1
$23$	\( T^{16} \) T^16
$29$	\( T^{16} \) T^16
$31$	\( T^{16} \) T^16
$37$	\( T^{16} \) T^16
$41$	\( T^{16} + 17 T^{14} + \cdots + 17 \) T^16 + 17T^14 + 119T^12 + 442T^10 + 935T^8 + 1122T^6 + 714T^4 + 204*T^2 + 17
$43$	\( T^{16} + 51 T^{9} + \cdots + 17 \) T^16 + 51T^9 - 238T^7 + 255T^5 + 17T^4 - 102T^3 + 85T^2 + 17*T + 17
$47$	\( T^{16} \) T^16
$53$	\( T^{16} \) T^16
$59$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 - 2T^11 - 4T^10 - 8T^9 - 16T^8 - 100T^7 + 21T^6 + 25T^5 + 50T^4 + 83T^3 + 47T^2 + 9*T + 1
$61$	\( T^{16} \) T^16
$67$	\( T^{16} + 51 T^{9} + \cdots + 17 \) T^16 + 51T^9 - 238T^7 + 255T^5 + 17T^4 - 102T^3 + 85T^2 + 17*T + 17
$71$	\( T^{16} \) T^16
$73$	\( T^{16} - 2 T^{15} + \cdots + 1 \) T^16 - 2T^15 + 4T^14 - 8T^13 - T^12 + 2T^11 - 4T^10 + 25T^9 + 35T^8 - 70T^7 + 140T^6 - 59T^5 - T^4 + 2T^3 + 30T^2 + 8*T + 1
$79$	\( T^{16} \) T^16
$83$	\( T^{16} + 17 T^{12} + \cdots + 17 \) T^16 + 17T^12 + 17T^9 + 85T^8 - 221T^5 + 119T^4 + 34T^2 + 68*T + 17
$89$	\( T^{16} + 51 T^{9} + \cdots + 17 \) T^16 + 51T^9 - 238T^7 + 255T^5 + 17T^4 - 102T^3 + 85T^2 + 17*T + 17
$97$	\( T^{16} - 17 T^{15} + \cdots + 17 \) T^16 - 17T^15 + 136T^14 - 680T^13 + 2380T^12 - 6188T^11 + 12376T^10 - 19448T^9 + 24310T^8 - 24310T^7 + 19448T^6 - 12376T^5 + 6188T^4 - 2380T^3 + 680T^2 - 136*T + 17
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