LMFDB - Newform orbit 2740.1.bs.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2740,1,Mod(19,2740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2740, base_ring=CyclotomicField(68))

chi = DirichletCharacter(H, H._module([34, 34, 23]))

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

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

chi := DirichletCharacter("2740.19");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2740 = 2^{2} \cdot 5 \cdot 137 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2740.bs (of order $68$, degree $32$, 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:	$1.36743813461$
Analytic rank:	$0$
Dimension:	$32$
Coefficient field:	$\Q(\zeta_{68})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} + \cdots + 1 $ x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{68}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{68} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2740\mathbb{Z}\right)^\times$.

$n$	$1097$	$1371$	$1921$
$\chi(n)$	$-1$	$-1$	$\zeta_{68}^{21}$

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} $

19.1

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

−0.932472

0.361242i

0.739009

−

0.673696i

0.273663

−

0.961826i

−0.445738

0.895163i

0.526432

−

0.850217i

0.0922684

0.995734i

39.1

0.273663

0.961826i

−0.850217

0.526432i

0.982973

−

0.183750i

−0.739009

−

0.673696i

−0.361242

−

0.932472i

0.445738

0.895163i

139.1

−0.739009

−

0.673696i

0.0922684

0.995734i

0.850217

−

0.526432i

0.602635

−

0.798017i

−0.895163

−

0.445738i

−0.982973

−

0.183750i

299.1

−0.0922684

0.995734i

−0.982973

−

0.183750i

−0.445738

−

0.895163i

0.273663

−

0.961826i

−0.798017

−

0.602635i

0.932472

−

0.361242i

379.1

0.850217

0.526432i

0.445738

0.895163i

−0.932472

−

0.361242i

−0.0922684

0.995734i

−0.673696

0.739009i

−0.602635

−

0.798017i

419.1

0.602635

−

0.798017i

−0.273663

−

0.961826i

−0.0922684

−

0.995734i

−0.932472

−

0.361242i

−0.183750

−

0.982973i

−0.850217

−

0.526432i

439.1

0.982973

0.183750i

0.932472

0.361242i

0.602635

−

0.798017i

0.850217

0.526432i

0.961826

−

0.273663i

0.739009

−

0.673696i

479.1

−0.739009

0.673696i

0.0922684

−

0.995734i

0.850217

0.526432i

0.602635

0.798017i

0.895163

−

0.445738i

−0.982973

0.183750i

539.1

−0.445738

0.895163i

−0.602635

−

0.798017i

−0.739009

0.673696i

0.982973

−

0.183750i

0.995734

0.0922684i

−0.273663

−

0.961826i

559.1

−0.0922684

−

0.995734i

−0.982973

0.183750i

−0.445738

0.895163i

0.273663

0.961826i

−0.798017

0.602635i

0.932472

0.361242i

839.1

0.602635

0.798017i

−0.273663

0.961826i

−0.0922684

0.995734i

−0.932472

0.361242i

0.183750

−

0.982973i

−0.850217

0.526432i

1079.1

0.602635

0.798017i

−0.273663

0.961826i

−0.0922684

0.995734i

−0.932472

0.361242i

−0.183750

0.982973i

−0.850217

0.526432i

1359.1

−0.0922684

−

0.995734i

−0.982973

0.183750i

−0.445738

0.895163i

0.273663

0.961826i

0.798017

−

0.602635i

0.932472

0.361242i

1379.1

−0.445738

0.895163i

−0.602635

−

0.798017i

−0.739009

0.673696i

0.982973

−

0.183750i

−0.995734

−

0.0922684i

−0.273663

−

0.961826i

1439.1

−0.739009

0.673696i

0.0922684

−

0.995734i

0.850217

0.526432i

0.602635

0.798017i

−0.895163

0.445738i

−0.982973

0.183750i

1479.1

0.982973

0.183750i

0.932472

0.361242i

0.602635

−

0.798017i

0.850217

0.526432i

−0.961826

0.273663i

0.739009

−

0.673696i

1499.1

0.602635

−

0.798017i

−0.273663

−

0.961826i

−0.0922684

−

0.995734i

−0.932472

−

0.361242i

0.183750

0.982973i

−0.850217

−

0.526432i

1539.1

0.850217

0.526432i

0.445738

0.895163i

−0.932472

−

0.361242i

−0.0922684

0.995734i

0.673696

−

0.739009i

−0.602635

−

0.798017i

1619.1

−0.0922684

0.995734i

−0.982973

−

0.183750i

−0.445738

−

0.895163i

0.273663

−

0.961826i

0.798017

0.602635i

0.932472

−

0.361242i

1779.1

−0.739009

−

0.673696i

0.0922684

0.995734i

0.850217

−

0.526432i

0.602635

−

0.798017i

0.895163

0.445738i

−0.982973

−

0.183750i

See all 32 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
685.s	even	68	1	inner
2740.bs	odd	68	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2740.1.bs.b	yes	32
4.b	odd	2	1	CM	2740.1.bs.b	yes	32
5.b	even	2	1		2740.1.bs.a	✓	32
20.d	odd	2	1		2740.1.bs.a	✓	32
137.g	even	68	1		2740.1.bs.a	✓	32
548.m	odd	68	1		2740.1.bs.a	✓	32
685.s	even	68	1	inner	2740.1.bs.b	yes	32
2740.bs	odd	68	1	inner	2740.1.bs.b	yes	32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2740.1.bs.a	✓	32	5.b	even	2	1
2740.1.bs.a	✓	32	20.d	odd	2	1
2740.1.bs.a	✓	32	137.g	even	68	1
2740.1.bs.a	✓	32	548.m	odd	68	1
2740.1.bs.b	yes	32	1.a	even	1	1	trivial
2740.1.bs.b	yes	32	4.b	odd	2	1	CM
2740.1.bs.b	yes	32	685.s	even	68	1	inner
2740.1.bs.b	yes	32	2740.bs	odd	68	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{13}^{32} - 2 T_{13}^{31} + 2 T_{13}^{30} - 4 T_{13}^{28} + 8 T_{13}^{27} - 8 T_{13}^{26} + 16 T_{13}^{24} + \cdots + 1 $ T13^32 - 2*T13^31 + 2*T13^30 - 4*T13^28 + 8*T13^27 - 8*T13^26 + 16*T13^24 - 32*T13^23 + 32*T13^22 - 64*T13^20 + 128*T13^19 - 128*T13^18 + 256*T13^16 - 510*T13^15 + 797*T13^14 + 6362*T13^13 + 51574*T13^12 + 213588*T13^11 + 469038*T13^10 + 633472*T13^9 + 573426*T13^8 + 364650*T13^7 + 166823*T13^6 + 55314*T13^5 + 13196*T13^4 + 2210*T13^3 + 247*T13^2 + 16*T13 + 1 acting on $S_{1}^{\mathrm{new}}(2740, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{16} - T^{15} + T^{14} + \cdots + 1)^{2} $ (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)^2
$3$	$ T^{32} $ T^32
$5$	$ (T^{16} - T^{15} + T^{14} + \cdots + 1)^{2} $ (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)^2
$7$	$ T^{32} $ T^32
$11$	$ T^{32} $ T^32
$13$	$ T^{32} - 2 T^{31} + \cdots + 1 $ T^32 - 2T^31 + 2T^30 - 4T^28 + 8T^27 - 8T^26 + 16T^24 - 32T^23 + 32T^22 - 64T^20 + 128T^19 - 128T^18 + 256T^16 - 510T^15 + 797T^14 + 6362T^13 + 51574T^12 + 213588T^11 + 469038T^10 + 633472T^9 + 573426T^8 + 364650T^7 + 166823T^6 + 55314T^5 + 13196T^4 + 2210T^3 + 247T^2 + 16*T + 1
$17$	$ T^{32} + 17 T^{30} + \cdots + 289 $ T^32 + 17T^30 + 136T^28 + 680T^26 + 2380T^24 + 6188T^22 + 12376T^20 + 19448T^18 + 24310T^16 + 24276T^14 + 20808T^12 + 45084T^8 - 46240T^6 + 25432T^4 - 4624T^2 + 289
$19$	$ T^{32} $ T^32
$23$	$ T^{32} $ T^32
$29$	$ T^{32} - 2 T^{31} + \cdots + 1 $ T^32 - 2T^31 + 2T^30 - 4T^28 + 8T^27 - 8T^26 + 170T^25 - 324T^24 + 308T^23 + 32T^22 - 680T^21 + 1483T^20 - 1606T^19 + 7403T^18 - 11594T^17 + 8637T^16 + 5916T^15 - 29106T^14 + 29618T^13 - 1024T^12 - 2278T^11 + 6604T^10 - 15044T^9 + 19651T^8 - 9180T^7 + 9692T^6 - 1024T^5 + 871T^4 + 204T^3 + 60T^2 - 18T + 1
$31$	$ T^{32} $ T^32
$37$	$ (T^{16} - 17 T^{14} + \cdots + 17)^{2} $ (T^16 - 17T^14 + 119T^12 - 442T^10 + 935T^8 - 1122T^6 + 714T^4 - 204*T^2 + 17)^2
$41$	$ T^{32} - 2 T^{31} + \cdots + 1 $ T^32 - 2T^31 + 2T^30 + 47T^28 - 94T^27 + 94T^26 + 815T^24 - 1630T^23 + 1630T^22 + 6294T^20 - 12588T^19 + 12588T^18 + 20231T^16 - 40460T^15 + 40458T^14 - 30T^13 + 19138T^12 - 38454T^11 + 38632T^10 + 528T^9 + 5490T^8 - 10166T^7 + 9352T^6 - 616T^5 - 132T^4 + 68T^3 + 128T^2 + 16*T + 1
$43$	$ T^{32} $ T^32
$47$	$ T^{32} $ T^32
$53$	$ T^{32} + 2 T^{31} + \cdots + 1 $ T^32 + 2T^31 + 2T^30 - 4T^28 - 8T^27 - 8T^26 + 16T^24 - 2T^23 - 36T^22 + 1360T^21 + 2792T^20 + 994T^19 - 3596T^18 - 8942T^17 - 10692T^16 - 3502T^15 + 14669T^14 + 36342T^13 + 60686T^12 + 48688T^11 + 59814T^10 + 22252T^9 + 24870T^8 + 5236T^7 + 6139T^6 + 1398T^5 + 72T^4 - 408T^3 + 43T^2 + 18*T + 1
$59$	$ T^{32} $ T^32
$61$	$ T^{32} - 4 T^{30} + \cdots + 1 $ T^32 - 4T^30 + 16T^28 + 21T^26 - 84T^24 + 370T^22 + 730T^20 - 2920T^18 + 5288T^16 - 2945T^14 + 12001T^12 + 6906T^10 + 3010T^8 - 4373T^6 + 730T^4 + 21*T^2 + 1
$67$	$ T^{32} $ T^32
$71$	$ T^{32} $ T^32
$73$	$ (T^{16} - 17 T^{11} + \cdots + 17)^{2} $ (T^16 - 17T^11 + 119T^8 + 68T^6 - 85T^5 + 221T^3 + 17T^2 - 34*T + 17)^2
$79$	$ T^{32} $ T^32
$83$	$ T^{32} $ T^32
$89$	$ T^{32} - 2 T^{31} + \cdots + 1 $ T^32 - 2T^31 + 2T^30 - 4T^28 + 8T^27 - 8T^26 + 16T^24 - 168T^23 + 1018T^22 - 1734T^21 + 1432T^20 + 604T^19 - 4072T^18 + 6936T^17 - 5728T^16 - 2414T^15 + 19752T^14 + 3472T^13 - 1075T^12 + 3876T^11 - 5381T^10 + 3010T^9 + 4742T^8 - 15504T^7 + 21677T^6 - 16834T^5 + 7331T^4 - 1802T^3 + 247T^2 - 18*T + 1
$97$	$ T^{32} - 2 T^{31} + \cdots + 65536 $ T^32 - 2T^31 + 2T^30 - 4T^28 + 8T^27 - 8T^26 + 16T^24 - 32T^23 + 32T^22 - 64T^20 + 128T^19 - 128T^18 + 256T^16 - 512T^14 + 1024T^13 - 1024T^12 + 2048T^10 - 4096T^9 + 4096T^8 - 8192T^6 + 16384T^5 - 16384T^4 + 32768T^2 - 65536*T + 65536
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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:	\( N \)	\(=\)	\( 2740 = 2^{2} \cdot 5 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2740.bs (of order \(68\), degree \(32\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{68}^{20} q^{2} - \zeta_{68}^{6} q^{4} + \zeta_{68}^{2} q^{5} + \zeta_{68}^{26} q^{8} - \zeta_{68}^{21} q^{9} +O(q^{10}) \) q - z^20 * q^2 - z^6 * q^4 + z^2 * q^5 + z^26 * q^8 - z^21 * q^9	\( q - \zeta_{68}^{20} q^{2} - \zeta_{68}^{6} q^{4} + \zeta_{68}^{2} q^{5} + \zeta_{68}^{26} q^{8} - \zeta_{68}^{21} q^{9} - \zeta_{68}^{22} q^{10} + (\zeta_{68}^{10} + \zeta_{68}^{5}) q^{13} + \zeta_{68}^{12} q^{16} + (\zeta_{68}^{33} - \zeta_{68}^{17}) q^{17} - \zeta_{68}^{7} q^{18} - \zeta_{68}^{8} q^{20} + \zeta_{68}^{4} q^{25} + ( - \zeta_{68}^{30} - \zeta_{68}^{25}) q^{26} + ( - \zeta_{68}^{28} + \zeta_{68}^{13}) q^{29} - \zeta_{68}^{32} q^{32} + (\zeta_{68}^{19} - \zeta_{68}^{3}) q^{34} + \zeta_{68}^{27} q^{36} + (\zeta_{68}^{31} - \zeta_{68}^{3}) q^{37} + \zeta_{68}^{28} q^{40} + ( - \zeta_{68}^{27} - \zeta_{68}^{24}) q^{41} - \zeta_{68}^{23} q^{45} + \zeta_{68}^{32} q^{49} - \zeta_{68}^{24} q^{50} + ( - \zeta_{68}^{16} - \zeta_{68}^{11}) q^{52} + (\zeta_{68}^{16} - \zeta_{68}^{15}) q^{53} + ( - \zeta_{68}^{33} - \zeta_{68}^{14}) q^{58} + (\zeta_{68}^{25} + \zeta_{68}) q^{61} - \zeta_{68}^{18} q^{64} + (\zeta_{68}^{12} + \zeta_{68}^{7}) q^{65} + (\zeta_{68}^{23} + \zeta_{68}^{5}) q^{68} + \zeta_{68}^{13} q^{72} + (\zeta_{68}^{22} - \zeta_{68}^{14}) q^{73} + (\zeta_{68}^{23} + \zeta_{68}^{17}) q^{74} + \zeta_{68}^{14} q^{80} - \zeta_{68}^{8} q^{81} + ( - \zeta_{68}^{13} - \zeta_{68}^{10}) q^{82} + ( - \zeta_{68}^{19} - \zeta_{68}) q^{85} + ( - \zeta_{68}^{29} - \zeta_{68}^{16}) q^{89} - \zeta_{68}^{9} q^{90} + ( - \zeta_{68}^{23} + \zeta_{68}^{6}) q^{97} + \zeta_{68}^{18} q^{98} +O(q^{100}) \) q - z^20 * q^2 - z^6 * q^4 + z^2 * q^5 + z^26 * q^8 - z^21 * q^9 - z^22 * q^10 + (z^10 + z^5) * q^13 + z^12 * q^16 + (z^33 - z^17) * q^17 - z^7 * q^18 - z^8 * q^20 + z^4 * q^25 + (-z^30 - z^25) * q^26 + (-z^28 + z^13) * q^29 - z^32 * q^32 + (z^19 - z^3) * q^34 + z^27 * q^36 + (z^31 - z^3) * q^37 + z^28 * q^40 + (-z^27 - z^24) * q^41 - z^23 * q^45 + z^32 * q^49 - z^24 * q^50 + (-z^16 - z^11) * q^52 + (z^16 - z^15) * q^53 + (-z^33 - z^14) * q^58 + (z^25 + z) * q^61 - z^18 * q^64 + (z^12 + z^7) * q^65 + (z^23 + z^5) * q^68 + z^13 * q^72 + (z^22 - z^14) * q^73 + (z^23 + z^17) * q^74 + z^14 * q^80 - z^8 * q^81 + (-z^13 - z^10) * q^82 + (-z^19 - z) * q^85 + (-z^29 - z^16) * q^89 - z^9 * q^90 + (-z^23 + z^6) * q^97 + z^18 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) \) 32 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^8	\( 32 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{8} - 2 q^{10} + 2 q^{13} - 2 q^{16} + 2 q^{20} - 2 q^{25} - 2 q^{26} + 2 q^{29} + 2 q^{32} - 2 q^{40} + 2 q^{41} - 2 q^{49} + 2 q^{50} + 2 q^{52} - 2 q^{53} - 2 q^{58} - 2 q^{64} - 2 q^{65} + 2 q^{80} + 2 q^{81} - 2 q^{82} + 2 q^{89} + 2 q^{97} + 2 q^{98}+O(q^{100}) \) 32 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^8 - 2 * q^10 + 2 * q^13 - 2 * q^16 + 2 * q^20 - 2 * q^25 - 2 * q^26 + 2 * q^29 + 2 * q^32 - 2 * q^40 + 2 * q^41 - 2 * q^49 + 2 * q^50 + 2 * q^52 - 2 * q^53 - 2 * q^58 - 2 * q^64 - 2 * q^65 + 2 * q^80 + 2 * q^81 - 2 * q^82 + 2 * q^89 + 2 * q^97 + 2 * q^98

Introduction

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Newspace parameters

Newform invariants

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Inner twists

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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	2740.1.bs.b

Level	$2740$
Weight	$1$
Character orbit	2740.bs
Analytic conductor	$1.367$
Analytic rank	$0$
Dimension	$32$
Projective image	$D_{68}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.36743813461\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Coefficient field:	\(\Q(\zeta_{68})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} + \cdots + 1 \) x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{68}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{68} - \cdots)\)

\(n\)	\(1097\)	\(1371\)	\(1921\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\zeta_{68}^{21}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 19.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
685.s	even	68	1	inner
2740.bs	odd	68	1	inner

$p$	$F_p(T)$
$2$	\( (T^{16} - T^{15} + T^{14} + \cdots + 1)^{2} \) (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)^2
$3$	\( T^{32} \) T^32
$5$	\( (T^{16} - T^{15} + T^{14} + \cdots + 1)^{2} \) (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)^2
$7$	\( T^{32} \) T^32
$11$	\( T^{32} \) T^32
$13$	\( T^{32} - 2 T^{31} + \cdots + 1 \) T^32 - 2T^31 + 2T^30 - 4T^28 + 8T^27 - 8T^26 + 16T^24 - 32T^23 + 32T^22 - 64T^20 + 128T^19 - 128T^18 + 256T^16 - 510T^15 + 797T^14 + 6362T^13 + 51574T^12 + 213588T^11 + 469038T^10 + 633472T^9 + 573426T^8 + 364650T^7 + 166823T^6 + 55314T^5 + 13196T^4 + 2210T^3 + 247T^2 + 16*T + 1
$17$	\( T^{32} + 17 T^{30} + \cdots + 289 \) T^32 + 17T^30 + 136T^28 + 680T^26 + 2380T^24 + 6188T^22 + 12376T^20 + 19448T^18 + 24310T^16 + 24276T^14 + 20808T^12 + 45084T^8 - 46240T^6 + 25432T^4 - 4624T^2 + 289
$19$	\( T^{32} \) T^32
$23$	\( T^{32} \) T^32
$29$	\( T^{32} - 2 T^{31} + \cdots + 1 \) T^32 - 2T^31 + 2T^30 - 4T^28 + 8T^27 - 8T^26 + 170T^25 - 324T^24 + 308T^23 + 32T^22 - 680T^21 + 1483T^20 - 1606T^19 + 7403T^18 - 11594T^17 + 8637T^16 + 5916T^15 - 29106T^14 + 29618T^13 - 1024T^12 - 2278T^11 + 6604T^10 - 15044T^9 + 19651T^8 - 9180T^7 + 9692T^6 - 1024T^5 + 871T^4 + 204T^3 + 60T^2 - 18T + 1
$31$	\( T^{32} \) T^32
$37$	\( (T^{16} - 17 T^{14} + \cdots + 17)^{2} \) (T^16 - 17T^14 + 119T^12 - 442T^10 + 935T^8 - 1122T^6 + 714T^4 - 204*T^2 + 17)^2
$41$	\( T^{32} - 2 T^{31} + \cdots + 1 \) T^32 - 2T^31 + 2T^30 + 47T^28 - 94T^27 + 94T^26 + 815T^24 - 1630T^23 + 1630T^22 + 6294T^20 - 12588T^19 + 12588T^18 + 20231T^16 - 40460T^15 + 40458T^14 - 30T^13 + 19138T^12 - 38454T^11 + 38632T^10 + 528T^9 + 5490T^8 - 10166T^7 + 9352T^6 - 616T^5 - 132T^4 + 68T^3 + 128T^2 + 16*T + 1
$43$	\( T^{32} \) T^32
$47$	\( T^{32} \) T^32
$53$	\( T^{32} + 2 T^{31} + \cdots + 1 \) T^32 + 2T^31 + 2T^30 - 4T^28 - 8T^27 - 8T^26 + 16T^24 - 2T^23 - 36T^22 + 1360T^21 + 2792T^20 + 994T^19 - 3596T^18 - 8942T^17 - 10692T^16 - 3502T^15 + 14669T^14 + 36342T^13 + 60686T^12 + 48688T^11 + 59814T^10 + 22252T^9 + 24870T^8 + 5236T^7 + 6139T^6 + 1398T^5 + 72T^4 - 408T^3 + 43T^2 + 18*T + 1
$59$	\( T^{32} \) T^32
$61$	\( T^{32} - 4 T^{30} + \cdots + 1 \) T^32 - 4T^30 + 16T^28 + 21T^26 - 84T^24 + 370T^22 + 730T^20 - 2920T^18 + 5288T^16 - 2945T^14 + 12001T^12 + 6906T^10 + 3010T^8 - 4373T^6 + 730T^4 + 21*T^2 + 1
$67$	\( T^{32} \) T^32
$71$	\( T^{32} \) T^32
$73$	\( (T^{16} - 17 T^{11} + \cdots + 17)^{2} \) (T^16 - 17T^11 + 119T^8 + 68T^6 - 85T^5 + 221T^3 + 17T^2 - 34*T + 17)^2
$79$	\( T^{32} \) T^32
$83$	\( T^{32} \) T^32
$89$	\( T^{32} - 2 T^{31} + \cdots + 1 \) T^32 - 2T^31 + 2T^30 - 4T^28 + 8T^27 - 8T^26 + 16T^24 - 168T^23 + 1018T^22 - 1734T^21 + 1432T^20 + 604T^19 - 4072T^18 + 6936T^17 - 5728T^16 - 2414T^15 + 19752T^14 + 3472T^13 - 1075T^12 + 3876T^11 - 5381T^10 + 3010T^9 + 4742T^8 - 15504T^7 + 21677T^6 - 16834T^5 + 7331T^4 - 1802T^3 + 247T^2 - 18*T + 1
$97$	\( T^{32} - 2 T^{31} + \cdots + 65536 \) T^32 - 2T^31 + 2T^30 - 4T^28 + 8T^27 - 8T^26 + 16T^24 - 32T^23 + 32T^22 - 64T^20 + 128T^19 - 128T^18 + 256T^16 - 512T^14 + 1024T^13 - 1024T^12 + 2048T^10 - 4096T^9 + 4096T^8 - 8192T^6 + 16384T^5 - 16384T^4 + 32768T^2 - 65536*T + 65536
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