LMFDB - Newform orbit 2541.1.bn.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2541,1,Mod(215,2541)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2541, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([15, 25, 27]))

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

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

chi := DirichletCharacter("2541.215");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2541 = 3 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2541.bn (of order $30$, degree $8$, not 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:	$1.26812419710$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$4$ over $\Q(\zeta_{30})$
Coefficient field:	$\Q(\zeta_{120})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 $ x^32 + x^28 - x^20 - x^16 - x^12 + x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{12}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

$q$-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_{120}^{22} q^{3} + \zeta_{120}^{28} q^{4} + \zeta_{120}^{43} q^{7} + \zeta_{120}^{44} q^{9}+O(q^{10}) $ q + z^22 * q^3 + z^28 * q^4 + z^43 * q^7 + z^44 * q^9	\( q + \zeta_{120}^{22} q^{3} + \zeta_{120}^{28} q^{4} + \zeta_{120}^{43} q^{7} + \zeta_{120}^{44} q^{9} + \zeta_{120}^{50} q^{12} + ( - \zeta_{120}^{29} - \zeta_{120}^{19}) q^{13} + \zeta_{120}^{56} q^{16} + (\zeta_{120}^{47} + \zeta_{120}^{17}) q^{19} - \zeta_{120}^{5} q^{21} + \zeta_{120}^{52} q^{25} - \zeta_{120}^{6} q^{27} - \zeta_{120}^{11} q^{28} + \zeta_{120}^{34} q^{31} - \zeta_{120}^{12} q^{36} + ( - \zeta_{120}^{58} + \zeta_{120}^{18}) q^{37} + ( - \zeta_{120}^{51} - \zeta_{120}^{41}) q^{39} + (\zeta_{120}^{55} + \zeta_{120}^{5}) q^{43} - \zeta_{120}^{18} q^{48} - \zeta_{120}^{26} q^{49} + ( - \zeta_{120}^{57} - \zeta_{120}^{47}) q^{52} + (\zeta_{120}^{39} - \zeta_{120}^{9}) q^{57} + (\zeta_{120}^{31} - \zeta_{120}^{21}) q^{61} - \zeta_{120}^{27} q^{63} - \zeta_{120}^{24} q^{64} + ( - \zeta_{120}^{53} - \zeta_{120}^{3}) q^{73} - \zeta_{120}^{14} q^{75} + (\zeta_{120}^{45} - \zeta_{120}^{15}) q^{76} + ( - \zeta_{120}^{59} + \zeta_{120}^{29}) q^{79} - \zeta_{120}^{28} q^{81} - \zeta_{120}^{33} q^{84} + (\zeta_{120}^{12} + \zeta_{120}^{2}) q^{91} + \zeta_{120}^{56} q^{93} + ( - \zeta_{120}^{44} + \zeta_{120}^{4}) q^{97} +O(q^{100}) \) q + z^22 * q^3 + z^28 * q^4 + z^43 * q^7 + z^44 * q^9 + z^50 * q^12 + (-z^29 - z^19) * q^13 + z^56 * q^16 + (z^47 + z^17) * q^19 - z^5 * q^21 + z^52 * q^25 - z^6 * q^27 - z^11 * q^28 + z^34 * q^31 - z^12 * q^36 + (-z^58 + z^18) * q^37 + (-z^51 - z^41) * q^39 + (z^55 + z^5) * q^43 - z^18 * q^48 - z^26 * q^49 + (-z^57 - z^47) * q^52 + (z^39 - z^9) * q^57 + (z^31 - z^21) * q^61 - z^27 * q^63 - z^24 * q^64 + (-z^53 - z^3) * q^73 - z^14 * q^75 + (z^45 - z^15) * q^76 + (-z^59 + z^29) * q^79 - z^28 * q^81 - z^33 * q^84 + (z^12 + z^2) * q^91 + z^56 * q^93 + (-z^44 + z^4) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 4 q^{4} - 4 q^{9}+O(q^{10}) $ 32 * q - 4 * q^4 - 4 * q^9	$ 32 q - 4 q^{4} - 4 q^{9} + 4 q^{16} - 4 q^{25} - 8 q^{36} + 8 q^{64} + 4 q^{81} + 8 q^{91} + 4 q^{93}+O(q^{100}) $ 32 * q - 4 * q^4 - 4 * q^9 + 4 * q^16 - 4 * q^25 - 8 * q^36 + 8 * q^64 + 4 * q^81 + 8 * q^91 + 4 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2541\mathbb{Z}\right)^\times$.

$n$	$848$	$1816$	$2059$
$\chi(n)$	$-1$	$\zeta_{120}^{20}$	$-\zeta_{120}^{48}$

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} $

215.1

−0.544639	+	0.838671i
0.544639	−	0.838671i
−0.838671	−	0.544639i
0.838671	+	0.544639i
0.933580	−	0.358368i
−0.933580	+	0.358368i
−0.358368	−	0.933580i
0.358368	+	0.933580i
0.0523360	−	0.998630i
−0.0523360	+	0.998630i
0.998630	+	0.0523360i
−0.998630	−	0.0523360i
0.777146	−	0.629320i
−0.777146	+	0.629320i
−0.629320	−	0.777146i
0.629320	+	0.777146i
0.777146	+	0.629320i
−0.777146	−	0.629320i
−0.629320	+	0.777146i
0.629320	−	0.777146i

−0.994522

−

0.104528i

−0.913545

−

0.406737i

−0.358368

−

0.933580i

0.978148

0.207912i

215.2

−0.994522

−

0.104528i

−0.913545

−

0.406737i

0.358368

0.933580i

0.978148

0.207912i

215.3

0.994522

0.104528i

−0.913545

−

0.406737i

−0.933580

0.358368i

0.978148

0.207912i

215.4

0.994522

0.104528i

−0.913545

−

0.406737i

0.933580

−

0.358368i

0.978148

0.207912i

887.1

−0.207912

−

0.978148i

−0.669131

0.743145i

−0.998630

0.0523360i

−0.913545

0.406737i

887.2

−0.207912

−

0.978148i

−0.669131

0.743145i

0.998630

−

0.0523360i

−0.913545

0.406737i

887.3

0.207912

0.978148i

−0.669131

0.743145i

−0.0523360

−

0.998630i

−0.913545

0.406737i

887.4

0.207912

0.978148i

−0.669131

0.743145i

0.0523360

0.998630i

−0.913545

0.406737i

941.1

−0.406737

−

0.913545i

0.104528

0.994522i

−0.777146

−

0.629320i

−0.669131

0.743145i

941.2

−0.406737

−

0.913545i

0.104528

0.994522i

0.777146

0.629320i

−0.669131

0.743145i

941.3

0.406737

0.913545i

0.104528

0.994522i

−0.629320

0.777146i

−0.669131

0.743145i

941.4

0.406737

0.913545i

0.104528

0.994522i

0.629320

−

0.777146i

−0.669131

0.743145i

1328.1

−0.743145

−

0.669131i

0.978148

−

0.207912i

−0.544639

0.838671i

0.104528

0.994522i

1328.2

−0.743145

−

0.669131i

0.978148

−

0.207912i

0.544639

−

0.838671i

0.104528

0.994522i

1328.3

0.743145

0.669131i

0.978148

−

0.207912i

−0.838671

−

0.544639i

0.104528

0.994522i

1328.4

0.743145

0.669131i

0.978148

−

0.207912i

0.838671

0.544639i

0.104528

0.994522i

1613.1

−0.743145

0.669131i

0.978148

0.207912i

−0.544639

−

0.838671i

0.104528

−

0.994522i

1613.2

−0.743145

0.669131i

0.978148

0.207912i

0.544639

0.838671i

0.104528

−

0.994522i

1613.3

0.743145

−

0.669131i

0.978148

0.207912i

−0.838671

0.544639i

0.104528

−

0.994522i

1613.4

0.743145

−

0.669131i

0.978148

0.207912i

0.838671

−

0.544639i

0.104528

−

0.994522i

See all 32 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
7.d	odd	6	1	inner
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner
21.g	even	6	1	inner
33.d	even	2	1	inner
33.f	even	10	3	inner
33.h	odd	10	3	inner
77.i	even	6	1	inner
77.n	even	30	3	inner
77.p	odd	30	3	inner
231.k	odd	6	1	inner
231.bc	even	30	3	inner
231.bf	odd	30	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2541.1.bn.a		32
3.b	odd	2	1	CM	2541.1.bn.a		32
7.d	odd	6	1	inner	2541.1.bn.a		32
11.b	odd	2	1	inner	2541.1.bn.a		32
11.c	even	5	1		2541.1.k.a	✓	8
11.c	even	5	3	inner	2541.1.bn.a		32
11.d	odd	10	1		2541.1.k.a	✓	8
11.d	odd	10	3	inner	2541.1.bn.a		32
21.g	even	6	1	inner	2541.1.bn.a		32
33.d	even	2	1	inner	2541.1.bn.a		32
33.f	even	10	1		2541.1.k.a	✓	8
33.f	even	10	3	inner	2541.1.bn.a		32
33.h	odd	10	1		2541.1.k.a	✓	8
33.h	odd	10	3	inner	2541.1.bn.a		32
77.i	even	6	1	inner	2541.1.bn.a		32
77.n	even	30	1		2541.1.k.a	✓	8
77.n	even	30	3	inner	2541.1.bn.a		32
77.p	odd	30	1		2541.1.k.a	✓	8
77.p	odd	30	3	inner	2541.1.bn.a		32
231.k	odd	6	1	inner	2541.1.bn.a		32
231.bc	even	30	1		2541.1.k.a	✓	8
231.bc	even	30	3	inner	2541.1.bn.a		32
231.bf	odd	30	1		2541.1.k.a	✓	8
231.bf	odd	30	3	inner	2541.1.bn.a		32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2541.1.k.a	✓	8	11.c	even	5	1
2541.1.k.a	✓	8	11.d	odd	10	1
2541.1.k.a	✓	8	33.f	even	10	1
2541.1.k.a	✓	8	33.h	odd	10	1
2541.1.k.a	✓	8	77.n	even	30	1
2541.1.k.a	✓	8	77.p	odd	30	1
2541.1.k.a	✓	8	231.bc	even	30	1
2541.1.k.a	✓	8	231.bf	odd	30	1
2541.1.bn.a		32	1.a	even	1	1	trivial
2541.1.bn.a		32	3.b	odd	2	1	CM
2541.1.bn.a		32	7.d	odd	6	1	inner
2541.1.bn.a		32	11.b	odd	2	1	inner
2541.1.bn.a		32	11.c	even	5	3	inner
2541.1.bn.a		32	11.d	odd	10	3	inner
2541.1.bn.a		32	21.g	even	6	1	inner
2541.1.bn.a		32	33.d	even	2	1	inner
2541.1.bn.a		32	33.f	even	10	3	inner
2541.1.bn.a		32	33.h	odd	10	3	inner
2541.1.bn.a		32	77.i	even	6	1	inner
2541.1.bn.a		32	77.n	even	30	3	inner
2541.1.bn.a		32	77.p	odd	30	3	inner
2541.1.bn.a		32	231.k	odd	6	1	inner
2541.1.bn.a		32	231.bc	even	30	3	inner
2541.1.bn.a		32	231.bf	odd	30	3	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2541, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{32} $ T^32
$3$	$ (T^{16} + T^{14} - T^{10} + \cdots + 1)^{2} $ (T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1)^2
$5$	$ T^{32} $ T^32
$7$	$ T^{32} + T^{28} + \cdots + 1 $ T^32 + T^28 - T^20 - T^16 - T^12 + T^4 + 1
$11$	$ T^{32} $ T^32
$13$	$ (T^{16} + 4 T^{14} + \cdots + 1)^{2} $ (T^16 + 4T^14 + 15T^12 + 56T^10 + 209T^8 + 56T^6 + 15T^4 + 4*T^2 + 1)^2
$17$	$ T^{32} $ T^32
$19$	$ (T^{16} - 2 T^{14} + \cdots + 256)^{2} $ (T^16 - 2T^14 + 8T^10 - 16T^8 + 32T^6 - 128*T^2 + 256)^2
$23$	$ T^{32} $ T^32
$29$	$ T^{32} $ T^32
$31$	$ (T^{16} + T^{14} - T^{10} + \cdots + 1)^{2} $ (T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1)^2
$37$	$ (T^{16} - 3 T^{14} + \cdots + 6561)^{2} $ (T^16 - 3T^14 + 27T^10 - 81T^8 + 243T^6 - 2187*T^2 + 6561)^2
$41$	$ T^{32} $ T^32
$43$	$ (T^{4} + 4 T^{2} + 1)^{8} $ (T^4 + 4*T^2 + 1)^8
$47$	$ T^{32} $ T^32
$53$	$ T^{32} $ T^32
$59$	$ T^{32} $ T^32
$61$	$ T^{32} - 4 T^{30} + \cdots + 1 $ T^32 - 4T^30 + T^28 + 52T^26 - 208T^24 + 776T^22 - 193T^20 - 10088T^18 + 40351T^16 - 10088T^14 - 193T^12 + 776T^10 - 208T^8 + 52T^6 + T^4 - 4*T^2 + 1
$67$	$ T^{32} $ T^32
$71$	$ T^{32} $ T^32
$73$	$ T^{32} - 4 T^{30} + \cdots + 1 $ T^32 - 4T^30 + T^28 + 52T^26 - 208T^24 + 776T^22 - 193T^20 - 10088T^18 + 40351T^16 - 10088T^14 - 193T^12 + 776T^10 - 208T^8 + 52T^6 + T^4 - 4*T^2 + 1
$79$	$ (T^{16} + 2 T^{14} + \cdots + 256)^{2} $ (T^16 + 2T^14 - 8T^10 - 16T^8 - 32T^6 + 128*T^2 + 256)^2
$83$	$ T^{32} $ T^32
$89$	$ T^{32} $ T^32
$97$	$ (T^{8} - 3 T^{6} + 9 T^{4} + \cdots + 81)^{4} $ (T^8 - 3T^6 + 9T^4 - 27*T^2 + 81)^4
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Overview	Random
Universe	Knowledge

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:	\( N \)	\(=\)	\( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2541.bn (of order \(30\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{120}^{22} q^{3} + \zeta_{120}^{28} q^{4} + \zeta_{120}^{43} q^{7} + \zeta_{120}^{44} q^{9}+O(q^{10}) \) q + z^22 * q^3 + z^28 * q^4 + z^43 * q^7 + z^44 * q^9	\( q + \zeta_{120}^{22} q^{3} + \zeta_{120}^{28} q^{4} + \zeta_{120}^{43} q^{7} + \zeta_{120}^{44} q^{9} + \zeta_{120}^{50} q^{12} + ( - \zeta_{120}^{29} - \zeta_{120}^{19}) q^{13} + \zeta_{120}^{56} q^{16} + (\zeta_{120}^{47} + \zeta_{120}^{17}) q^{19} - \zeta_{120}^{5} q^{21} + \zeta_{120}^{52} q^{25} - \zeta_{120}^{6} q^{27} - \zeta_{120}^{11} q^{28} + \zeta_{120}^{34} q^{31} - \zeta_{120}^{12} q^{36} + ( - \zeta_{120}^{58} + \zeta_{120}^{18}) q^{37} + ( - \zeta_{120}^{51} - \zeta_{120}^{41}) q^{39} + (\zeta_{120}^{55} + \zeta_{120}^{5}) q^{43} - \zeta_{120}^{18} q^{48} - \zeta_{120}^{26} q^{49} + ( - \zeta_{120}^{57} - \zeta_{120}^{47}) q^{52} + (\zeta_{120}^{39} - \zeta_{120}^{9}) q^{57} + (\zeta_{120}^{31} - \zeta_{120}^{21}) q^{61} - \zeta_{120}^{27} q^{63} - \zeta_{120}^{24} q^{64} + ( - \zeta_{120}^{53} - \zeta_{120}^{3}) q^{73} - \zeta_{120}^{14} q^{75} + (\zeta_{120}^{45} - \zeta_{120}^{15}) q^{76} + ( - \zeta_{120}^{59} + \zeta_{120}^{29}) q^{79} - \zeta_{120}^{28} q^{81} - \zeta_{120}^{33} q^{84} + (\zeta_{120}^{12} + \zeta_{120}^{2}) q^{91} + \zeta_{120}^{56} q^{93} + ( - \zeta_{120}^{44} + \zeta_{120}^{4}) q^{97} +O(q^{100}) \) q + z^22 * q^3 + z^28 * q^4 + z^43 * q^7 + z^44 * q^9 + z^50 * q^12 + (-z^29 - z^19) * q^13 + z^56 * q^16 + (z^47 + z^17) * q^19 - z^5 * q^21 + z^52 * q^25 - z^6 * q^27 - z^11 * q^28 + z^34 * q^31 - z^12 * q^36 + (-z^58 + z^18) * q^37 + (-z^51 - z^41) * q^39 + (z^55 + z^5) * q^43 - z^18 * q^48 - z^26 * q^49 + (-z^57 - z^47) * q^52 + (z^39 - z^9) * q^57 + (z^31 - z^21) * q^61 - z^27 * q^63 - z^24 * q^64 + (-z^53 - z^3) * q^73 - z^14 * q^75 + (z^45 - z^15) * q^76 + (-z^59 + z^29) * q^79 - z^28 * q^81 - z^33 * q^84 + (z^12 + z^2) * q^91 + z^56 * q^93 + (-z^44 + z^4) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 4 q^{4} - 4 q^{9}+O(q^{10}) \) 32 * q - 4 * q^4 - 4 * q^9	\( 32 q - 4 q^{4} - 4 q^{9} + 4 q^{16} - 4 q^{25} - 8 q^{36} + 8 q^{64} + 4 q^{81} + 8 q^{91} + 4 q^{93}+O(q^{100}) \) 32 * q - 4 * q^4 - 4 * q^9 + 4 * q^16 - 4 * q^25 - 8 * q^36 + 8 * q^64 + 4 * q^81 + 8 * q^91 + 4 * q^93

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	2541.1.bn.a

Level	$2541$
Weight	$1$
Character orbit	2541.bn
Analytic conductor	$1.268$
Analytic rank	$0$
Dimension	$32$
Projective image	$D_{12}$
CM discriminant	-3
Inner twists	$32$

Self dual:	no
Analytic conductor:	\(1.26812419710\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{30})\)
Coefficient field:	\(\Q(\zeta_{120})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 \) x^32 + x^28 - x^20 - x^16 - x^12 + x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{12}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

\(n\)	\(848\)	\(1816\)	\(2059\)
\(\chi(n)\)	\(-1\)	\(\zeta_{120}^{20}\)	\(-\zeta_{120}^{48}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 215.4
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
7.d	odd	6	1	inner
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner
21.g	even	6	1	inner
33.d	even	2	1	inner
33.f	even	10	3	inner
33.h	odd	10	3	inner
77.i	even	6	1	inner
77.n	even	30	3	inner
77.p	odd	30	3	inner
231.k	odd	6	1	inner
231.bc	even	30	3	inner
231.bf	odd	30	3	inner

$p$	$F_p(T)$
$2$	\( T^{32} \) T^32
$3$	\( (T^{16} + T^{14} - T^{10} + \cdots + 1)^{2} \) (T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1)^2
$5$	\( T^{32} \) T^32
$7$	\( T^{32} + T^{28} + \cdots + 1 \) T^32 + T^28 - T^20 - T^16 - T^12 + T^4 + 1
$11$	\( T^{32} \) T^32
$13$	\( (T^{16} + 4 T^{14} + \cdots + 1)^{2} \) (T^16 + 4T^14 + 15T^12 + 56T^10 + 209T^8 + 56T^6 + 15T^4 + 4*T^2 + 1)^2
$17$	\( T^{32} \) T^32
$19$	\( (T^{16} - 2 T^{14} + \cdots + 256)^{2} \) (T^16 - 2T^14 + 8T^10 - 16T^8 + 32T^6 - 128*T^2 + 256)^2
$23$	\( T^{32} \) T^32
$29$	\( T^{32} \) T^32
$31$	\( (T^{16} + T^{14} - T^{10} + \cdots + 1)^{2} \) (T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1)^2
$37$	\( (T^{16} - 3 T^{14} + \cdots + 6561)^{2} \) (T^16 - 3T^14 + 27T^10 - 81T^8 + 243T^6 - 2187*T^2 + 6561)^2
$41$	\( T^{32} \) T^32
$43$	\( (T^{4} + 4 T^{2} + 1)^{8} \) (T^4 + 4*T^2 + 1)^8
$47$	\( T^{32} \) T^32
$53$	\( T^{32} \) T^32
$59$	\( T^{32} \) T^32
$61$	\( T^{32} - 4 T^{30} + \cdots + 1 \) T^32 - 4T^30 + T^28 + 52T^26 - 208T^24 + 776T^22 - 193T^20 - 10088T^18 + 40351T^16 - 10088T^14 - 193T^12 + 776T^10 - 208T^8 + 52T^6 + T^4 - 4*T^2 + 1
$67$	\( T^{32} \) T^32
$71$	\( T^{32} \) T^32
$73$	\( T^{32} - 4 T^{30} + \cdots + 1 \) T^32 - 4T^30 + T^28 + 52T^26 - 208T^24 + 776T^22 - 193T^20 - 10088T^18 + 40351T^16 - 10088T^14 - 193T^12 + 776T^10 - 208T^8 + 52T^6 + T^4 - 4*T^2 + 1
$79$	\( (T^{16} + 2 T^{14} + \cdots + 256)^{2} \) (T^16 + 2T^14 - 8T^10 - 16T^8 - 32T^6 + 128*T^2 + 256)^2
$83$	\( T^{32} \) T^32
$89$	\( T^{32} \) T^32
$97$	\( (T^{8} - 3 T^{6} + 9 T^{4} + \cdots + 81)^{4} \) (T^8 - 3T^6 + 9T^4 - 27*T^2 + 81)^4
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