LMFDB - Newform orbit 2120.1.bx.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2120,1,Mod(99,2120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2120, base_ring=CyclotomicField(26))

chi = DirichletCharacter(H, H._module([13, 13, 13, 20]))

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

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

chi := DirichletCharacter("2120.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2120 = 2^{3} \cdot 5 \cdot 53 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2120.bx (of order $26$, degree $12$, 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:	$1.05801782678$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{26})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ 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:	$D_{13}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{13} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2120\mathbb{Z}\right)^\times$.

$n$	$161$	$1061$	$1591$	$1697$
$\chi(n)$	$\zeta_{26}^{6}$	$-1$	$-1$	$-1$

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} $

99.1

0.970942	−	0.239316i
0.354605	−	0.935016i
−0.885456	+	0.464723i
0.748511	+	0.663123i
−0.568065	−	0.822984i
−0.120537	−	0.992709i
−0.885456	−	0.464723i
0.748511	−	0.663123i
0.970942	+	0.239316i
−0.120537	+	0.992709i
−0.568065	+	0.822984i
0.354605	+	0.935016i

0.748511

−

0.663123i

0.120537

−

0.992709i

−0.885456

−

0.464723i

1.32555

−

1.17433i

−0.568065

−

0.822984i

0.885456

0.464723i

−0.970942

0.239316i

259.1

−0.885456

0.464723i

0.568065

−

0.822984i

0.748511

−

0.663123i

1.32555

−

0.695701i

−0.120537

0.992709i

−0.748511

0.663123i

−0.354605

0.935016i

579.1

−0.120537

0.992709i

−0.970942

−

0.239316i

−0.568065

−

0.822984i

−0.136945

1.12785i

0.354605

−

0.935016i

0.568065

0.822984i

0.885456

−

0.464723i

619.1

−0.568065

0.822984i

−0.354605

−

0.935016i

−0.120537

0.992709i

−0.136945

0.198399i

0.970942

0.239316i

0.120537

−

0.992709i

−0.748511

−

0.663123i

699.1

0.970942

−

0.239316i

0.885456

−

0.464723i

0.354605

0.935016i

−0.688601

0.169725i

0.748511

−

0.663123i

−0.354605

−

0.935016i

0.568065

0.822984i

819.1

0.354605

0.935016i

−0.748511

0.663123i

0.970942

0.239316i

−0.688601

−

1.81569i

−0.885456

−

0.464723i

−0.970942

−

0.239316i

0.120537

0.992709i

1179.1

−0.120537

−

0.992709i

−0.970942

0.239316i

−0.568065

0.822984i

−0.136945

−

1.12785i

0.354605

0.935016i

0.568065

−

0.822984i

0.885456

0.464723i

1459.1

−0.568065

−

0.822984i

−0.354605

0.935016i

−0.120537

−

0.992709i

−0.136945

−

0.198399i

0.970942

−

0.239316i

0.120537

0.992709i

−0.748511

0.663123i

1499.1

0.748511

0.663123i

0.120537

0.992709i

−0.885456

0.464723i

1.32555

1.17433i

−0.568065

0.822984i

0.885456

−

0.464723i

−0.970942

−

0.239316i

1579.1

0.354605

−

0.935016i

−0.748511

−

0.663123i

0.970942

−

0.239316i

−0.688601

1.81569i

−0.885456

0.464723i

−0.970942

0.239316i

0.120537

−

0.992709i

1659.1

0.970942

0.239316i

0.885456

0.464723i

0.354605

−

0.935016i

−0.688601

−

0.169725i

0.748511

0.663123i

−0.354605

0.935016i

0.568065

−

0.822984i

1899.1

−0.885456

−

0.464723i

0.568065

0.822984i

0.748511

0.663123i

1.32555

0.695701i

−0.120537

−

0.992709i

−0.748511

−

0.663123i

−0.354605

−

0.935016i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
40.e	odd	2	1	CM by $\Q(\sqrt{-10}) $
53.d	even	13	1	inner
2120.bx	odd	26	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2120.1.bx.b	yes	12
5.b	even	2	1		2120.1.bx.a	✓	12
8.d	odd	2	1		2120.1.bx.a	✓	12
40.e	odd	2	1	CM	2120.1.bx.b	yes	12
53.d	even	13	1	inner	2120.1.bx.b	yes	12
265.n	even	26	1		2120.1.bx.a	✓	12
424.o	odd	26	1		2120.1.bx.a	✓	12
2120.bx	odd	26	1	inner	2120.1.bx.b	yes	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2120.1.bx.a	✓	12	5.b	even	2	1
2120.1.bx.a	✓	12	8.d	odd	2	1
2120.1.bx.a	✓	12	265.n	even	26	1
2120.1.bx.a	✓	12	424.o	odd	26	1
2120.1.bx.b	yes	12	1.a	even	1	1	trivial
2120.1.bx.b	yes	12	40.e	odd	2	1	CM
2120.1.bx.b	yes	12	53.d	even	13	1	inner
2120.1.bx.b	yes	12	2120.bx	odd	26	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{12} - 2 T_{7}^{11} + 4 T_{7}^{10} - 8 T_{7}^{9} + 16 T_{7}^{8} - 6 T_{7}^{7} - T_{7}^{6} + \cdots + 1 $ T7^12 - 2*T7^11 + 4*T7^10 - 8*T7^9 + 16*T7^8 - 6*T7^7 - T7^6 + 2*T7^5 - 4*T7^4 + 21*T7^3 + 23*T7^2 + 6*T7 + 1 acting on $S_{1}^{\mathrm{new}}(2120, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - T^{11} + \cdots + 1 $ 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^{12} $ T^12
$5$	$ T^{12} - T^{11} + \cdots + 1 $ T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$7$	$ T^{12} - 2 T^{11} + \cdots + 1 $ T^12 - 2T^11 + 4T^10 - 8T^9 + 16T^8 - 6T^7 - T^6 + 2T^5 - 4T^4 + 21T^3 + 23T^2 + 6T + 1
$11$	$ T^{12} + 2 T^{11} + \cdots + 1 $ T^12 + 2T^11 + 4T^10 + 8T^9 + 3T^8 + 6T^7 + 12T^6 - 15T^5 + 9T^4 + 18T^3 + 23T^2 + 7*T + 1
$13$	$ T^{12} - 2 T^{11} + \cdots + 1 $ T^12 - 2T^11 + 4T^10 - 8T^9 + 16T^8 - 19T^7 + 38T^6 - 11T^5 + 22T^4 + 8T^3 + 10T^2 - 7*T + 1
$17$	$ T^{12} $ T^12
$19$	$ T^{12} + 2 T^{11} + \cdots + 1 $ T^12 + 2T^11 + 4T^10 + 8T^9 + 16T^8 + 6T^7 - T^6 - 2T^5 - 4T^4 - 21T^3 + 23T^2 - 6T + 1
$23$	$ (T^{6} - T^{5} - 5 T^{4} + \cdots - 1)^{2} $ (T^6 - T^5 - 5T^4 + 4T^3 + 6T^2 - 3T - 1)^2
$29$	$ T^{12} $ T^12
$31$	$ T^{12} $ T^12
$37$	$ T^{12} - 2 T^{11} + \cdots + 1 $ T^12 - 2T^11 + 4T^10 - 8T^9 + 16T^8 - 6T^7 - T^6 + 2T^5 - 4T^4 + 21T^3 + 23T^2 + 6T + 1
$41$	$ T^{12} + 2 T^{11} + \cdots + 1 $ T^12 + 2T^11 + 4T^10 + 8T^9 + 16T^8 + 32T^7 + 64T^6 + 115T^5 + 139T^4 + 96T^3 + 36T^2 + 7*T + 1
$43$	$ T^{12} $ T^12
$47$	$ T^{12} - 2 T^{11} + \cdots + 1 $ T^12 - 2T^11 + 4T^10 - 8T^9 + 16T^8 - 19T^7 + 38T^6 - 11T^5 + 22T^4 + 8T^3 + 10T^2 - 7*T + 1
$53$	$ T^{12} - T^{11} + \cdots + 1 $ T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$59$	$ T^{12} + 2 T^{11} + \cdots + 1 $ T^12 + 2T^11 + 4T^10 + 8T^9 + 16T^8 + 32T^7 + 64T^6 + 115T^5 + 139T^4 + 96T^3 + 36T^2 + 7*T + 1
$61$	$ T^{12} $ T^12
$67$	$ T^{12} $ T^12
$71$	$ T^{12} $ T^12
$73$	$ T^{12} $ T^12
$79$	$ T^{12} $ T^12
$83$	$ T^{12} $ T^12
$89$	$ T^{12} + 2 T^{11} + \cdots + 1 $ T^12 + 2T^11 + 4T^10 - 5T^9 - 10T^8 - 20T^7 + 12T^6 + 24T^5 + 35T^4 + 5T^3 + 10T^2 - 6*T + 1
$97$	$ T^{12} $ T^12
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Overview	Random
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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:	\( N \)	\(=\)	\( 2120 = 2^{3} \cdot 5 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2120.bx (of order \(26\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{26}^{3} q^{2} + \zeta_{26}^{6} q^{4} + \zeta_{26}^{11} q^{5} + (\zeta_{26}^{5} + \zeta_{26}) q^{7} + \zeta_{26}^{9} q^{8} - \zeta_{26}^{11} q^{9} +O(q^{10}) \) q + z^3 * q^2 + z^6 * q^4 + z^11 * q^5 + (z^5 + z) * q^7 + z^9 * q^8 - z^11 * q^9	\( q + \zeta_{26}^{3} q^{2} + \zeta_{26}^{6} q^{4} + \zeta_{26}^{11} q^{5} + (\zeta_{26}^{5} + \zeta_{26}) q^{7} + \zeta_{26}^{9} q^{8} - \zeta_{26}^{11} q^{9} - \zeta_{26} q^{10} + (\zeta_{26}^{8} + \zeta_{26}^{2}) q^{11} + ( - \zeta_{26}^{10} - \zeta_{26}^{4}) q^{13} + (\zeta_{26}^{8} + \zeta_{26}^{4}) q^{14} + \zeta_{26}^{12} q^{16} + \zeta_{26} q^{18} + ( - \zeta_{26}^{11} - \zeta_{26}^{3}) q^{19} - \zeta_{26}^{4} q^{20} + (\zeta_{26}^{11} + \zeta_{26}^{5}) q^{22} + (\zeta_{26}^{11} - \zeta_{26}^{2}) q^{23} - \zeta_{26}^{9} q^{25} + ( - \zeta_{26}^{7} + 1) q^{26} + (\zeta_{26}^{11} + \zeta_{26}^{7}) q^{28} - \zeta_{26}^{2} q^{32} + (\zeta_{26}^{12} - \zeta_{26}^{3}) q^{35} + \zeta_{26}^{4} q^{36} + (\zeta_{26}^{7} - \zeta_{26}^{4}) q^{37} + ( - \zeta_{26}^{6} + \zeta_{26}) q^{38} - \zeta_{26}^{7} q^{40} + (\zeta_{26}^{12} - \zeta_{26}^{11}) q^{41} + (\zeta_{26}^{8} - \zeta_{26}) q^{44} + \zeta_{26}^{9} q^{45} + ( - \zeta_{26}^{5} - \zeta_{26}) q^{46} + (\zeta_{26}^{3} + \zeta_{26}) q^{47} + (\zeta_{26}^{10} + \cdots + \zeta_{26}^{2}) q^{49} + \cdots + (\zeta_{26}^{6} + 1) q^{99} +O(q^{100}) \) q + z^3 * q^2 + z^6 * q^4 + z^11 * q^5 + (z^5 + z) * q^7 + z^9 * q^8 - z^11 * q^9 - z * q^10 + (z^8 + z^2) * q^11 + (-z^10 - z^4) * q^13 + (z^8 + z^4) * q^14 + z^12 * q^16 + z * q^18 + (-z^11 - z^3) * q^19 - z^4 * q^20 + (z^11 + z^5) * q^22 + (z^11 - z^2) * q^23 - z^9 * q^25 + (-z^7 + 1) * q^26 + (z^11 + z^7) * q^28 - z^2 * q^32 + (z^12 - z^3) * q^35 + z^4 * q^36 + (z^7 - z^4) * q^37 + (-z^6 + z) * q^38 - z^7 * q^40 + (z^12 - z^11) * q^41 + (z^8 - z) * q^44 + z^9 * q^45 + (-z^5 - z) * q^46 + (z^3 + z) * q^47 + (z^10 + z^6 + z^2) * q^49 - z^12 * q^50 + (-z^10 + z^3) * q^52 - z^2 * q^53 + (-z^6 - 1) * q^55 + (z^10 - z) * q^56 + (z^10 - z^7) * q^59 + (-z^12 + z^3) * q^63 - z^5 * q^64 + (z^8 + z^2) * q^65 + (-z^6 - z^2) * q^70 + z^7 * q^72 + (z^10 - z^7) * q^74 + (-z^9 + z^4) * q^76 + (z^9 + z^7 + z^3 - 1) * q^77 - z^10 * q^80 - z^9 * q^81 + (-z^2 + z) * q^82 + (z^11 - z^4) * q^88 + (-z^5 - z^3) * q^89 + z^12 * q^90 + (-z^11 - z^9 - z^5 + z^2) * q^91 + (-z^8 - z^4) * q^92 + (z^6 + z^4) * q^94 + (z^9 + z) * q^95 + (z^9 + z^5 - 1) * q^98 + (z^6 + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} - q^{4} + q^{5} + 2 q^{7} + q^{8} - q^{9}+O(q^{10}) \) 12 * q + q^2 - q^4 + q^5 + 2 * q^7 + q^8 - q^9	\( 12 q + q^{2} - q^{4} + q^{5} + 2 q^{7} + q^{8} - q^{9} - q^{10} - 2 q^{11} + 2 q^{13} - 2 q^{14} - q^{16} + q^{18} - 2 q^{19} + q^{20} + 2 q^{22} + 2 q^{23} - q^{25} + 11 q^{26} + 2 q^{28} + q^{32} - 2 q^{35} - q^{36} + 2 q^{37} + 2 q^{38} - q^{40} - 2 q^{41} - 2 q^{44} + q^{45} - 2 q^{46} + 2 q^{47} - 3 q^{49} + q^{50} + 2 q^{52} + q^{53} - 11 q^{55} - 2 q^{56} - 2 q^{59} + 2 q^{63} - q^{64} - 2 q^{65} + 2 q^{70} + q^{72} - 2 q^{74} - 2 q^{76} - 9 q^{77} + q^{80} - q^{81} + 2 q^{82} + 2 q^{88} - 2 q^{89} - q^{90} - 4 q^{91} + 2 q^{92} - 2 q^{94} + 2 q^{95} - 10 q^{98} + 11 q^{99}+O(q^{100}) \) 12 * q + q^2 - q^4 + q^5 + 2 * q^7 + q^8 - q^9 - q^10 - 2 * q^11 + 2 * q^13 - 2 * q^14 - q^16 + q^18 - 2 * q^19 + q^20 + 2 * q^22 + 2 * q^23 - q^25 + 11 * q^26 + 2 * q^28 + q^32 - 2 * q^35 - q^36 + 2 * q^37 + 2 * q^38 - q^40 - 2 * q^41 - 2 * q^44 + q^45 - 2 * q^46 + 2 * q^47 - 3 * q^49 + q^50 + 2 * q^52 + q^53 - 11 * q^55 - 2 * q^56 - 2 * q^59 + 2 * q^63 - q^64 - 2 * q^65 + 2 * q^70 + q^72 - 2 * q^74 - 2 * q^76 - 9 * q^77 + q^80 - q^81 + 2 * q^82 + 2 * q^88 - 2 * q^89 - q^90 - 4 * q^91 + 2 * q^92 - 2 * q^94 + 2 * q^95 - 10 * q^98 + 11 * 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	2120.1.bx.b

Level	$2120$
Weight	$1$
Character orbit	2120.bx
Analytic conductor	$1.058$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{13}$
CM discriminant	-40
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.05801782678\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\Q(\zeta_{26})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) 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:	\(D_{13}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{13} - \cdots)\)

\(n\)	\(161\)	\(1061\)	\(1591\)	\(1697\)
\(\chi(n)\)	\(\zeta_{26}^{6}\)	\(-1\)	\(-1\)	\(-1\)

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

$p$	$F_p(T)$
$2$	\( T^{12} - T^{11} + \cdots + 1 \) T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$	\( T^{12} \) T^12
$5$	\( T^{12} - T^{11} + \cdots + 1 \) T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$7$	\( T^{12} - 2 T^{11} + \cdots + 1 \) T^12 - 2T^11 + 4T^10 - 8T^9 + 16T^8 - 6T^7 - T^6 + 2T^5 - 4T^4 + 21T^3 + 23T^2 + 6T + 1
$11$	\( T^{12} + 2 T^{11} + \cdots + 1 \) T^12 + 2T^11 + 4T^10 + 8T^9 + 3T^8 + 6T^7 + 12T^6 - 15T^5 + 9T^4 + 18T^3 + 23T^2 + 7*T + 1
$13$	\( T^{12} - 2 T^{11} + \cdots + 1 \) T^12 - 2T^11 + 4T^10 - 8T^9 + 16T^8 - 19T^7 + 38T^6 - 11T^5 + 22T^4 + 8T^3 + 10T^2 - 7*T + 1
$17$	\( T^{12} \) T^12
$19$	\( T^{12} + 2 T^{11} + \cdots + 1 \) T^12 + 2T^11 + 4T^10 + 8T^9 + 16T^8 + 6T^7 - T^6 - 2T^5 - 4T^4 - 21T^3 + 23T^2 - 6T + 1
$23$	\( (T^{6} - T^{5} - 5 T^{4} + \cdots - 1)^{2} \) (T^6 - T^5 - 5T^4 + 4T^3 + 6T^2 - 3T - 1)^2
$29$	\( T^{12} \) T^12
$31$	\( T^{12} \) T^12
$37$	\( T^{12} - 2 T^{11} + \cdots + 1 \) T^12 - 2T^11 + 4T^10 - 8T^9 + 16T^8 - 6T^7 - T^6 + 2T^5 - 4T^4 + 21T^3 + 23T^2 + 6T + 1
$41$	\( T^{12} + 2 T^{11} + \cdots + 1 \) T^12 + 2T^11 + 4T^10 + 8T^9 + 16T^8 + 32T^7 + 64T^6 + 115T^5 + 139T^4 + 96T^3 + 36T^2 + 7*T + 1
$43$	\( T^{12} \) T^12
$47$	\( T^{12} - 2 T^{11} + \cdots + 1 \) T^12 - 2T^11 + 4T^10 - 8T^9 + 16T^8 - 19T^7 + 38T^6 - 11T^5 + 22T^4 + 8T^3 + 10T^2 - 7*T + 1
$53$	\( T^{12} - T^{11} + \cdots + 1 \) T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$59$	\( T^{12} + 2 T^{11} + \cdots + 1 \) T^12 + 2T^11 + 4T^10 + 8T^9 + 16T^8 + 32T^7 + 64T^6 + 115T^5 + 139T^4 + 96T^3 + 36T^2 + 7*T + 1
$61$	\( T^{12} \) T^12
$67$	\( T^{12} \) T^12
$71$	\( T^{12} \) T^12
$73$	\( T^{12} \) T^12
$79$	\( T^{12} \) T^12
$83$	\( T^{12} \) T^12
$89$	\( T^{12} + 2 T^{11} + \cdots + 1 \) T^12 + 2T^11 + 4T^10 - 5T^9 - 10T^8 - 20T^7 + 12T^6 + 24T^5 + 35T^4 + 5T^3 + 10T^2 - 6*T + 1
$97$	\( T^{12} \) T^12
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