LMFDB - Newform orbit 3267.1.be.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3267,1,Mod(245,3267)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3267, base_ring=CyclotomicField(90))

chi = DirichletCharacter(H, H._module([5, 72]))

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

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

chi := DirichletCharacter("3267.245");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3267 = 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3267.be (of order $90$, degree $24$, 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:	$1.63044539627$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{45})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 $ x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{18}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{18} - \cdots)$

$q$-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_{90}^{11} q^{3} + \zeta_{90}^{14} q^{4} + (\zeta_{90}^{13} - \zeta_{90}^{3}) q^{5} + \zeta_{90}^{22} q^{9} + \zeta_{90}^{25} q^{12} + (\zeta_{90}^{24} - \zeta_{90}^{14}) q^{15} + \zeta_{90}^{28} q^{16}+ \cdots + ( - \zeta_{90}^{27} + \zeta_{90}^{2}) q^{97}+O(q^{100}) $ q + z^11 * q^3 + z^14 * q^4 + (z^13 - z^3) * q^5 + z^22 * q^9 + z^25 * q^12 + (z^24 - z^14) * q^15 + z^28 * q^16 + (z^27 - z^17) * q^20 + (-z^35 - z^20) * q^23 + (z^26 - z^16 + z^6) * q^25 + z^33 * q^27 + (-z^12 + z^7) * q^31 + z^36 * q^36 + (z^29 + z^19) * q^37 + (z^35 - z^25) * q^45 + (-z^41 + z^21) * q^47 + z^39 * q^48 - z^8 * q^49 + (-z^7 - z^2) * q^53 + (-z^24 - z^19) * q^59 + (z^38 - z^28) * q^60 + z^42 * q^64 + (-z^40 - 1) * q^67 + (-z^31 + z) * q^69 + (-z^43 + z^23) * q^71 + (z^37 - z^27 + z^17) * q^75 + (z^41 - z^31) * q^80 + z^44 * q^81 + (-z^15 - 1) * q^89 + (-z^34 + z^4) * q^92 + (-z^23 + z^18) * q^93 + (-z^27 + z^2) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 3 q^{5} + 3 q^{15} + 6 q^{20} + 3 q^{25} - 3 q^{27} - 3 q^{31} - 6 q^{36} - 3 q^{47} - 3 q^{48} - 3 q^{59} + 3 q^{64} - 24 q^{67} - 6 q^{75} - 36 q^{89} - 6 q^{93} - 6 q^{97}+O(q^{100}) $ 24 * q + 3 * q^5 + 3 * q^15 + 6 * q^20 + 3 * q^25 - 3 * q^27 - 3 * q^31 - 6 * q^36 - 3 * q^47 - 3 * q^48 - 3 * q^59 + 3 * q^64 - 24 * q^67 - 6 * q^75 - 36 * q^89 - 6 * q^93 - 6 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times$.

$n$	$244$	$3026$
$\chi(n)$	$-\zeta_{90}^{9}$	$\zeta_{90}^{5}$

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} $

245.1

−0.374607	+	0.927184i
0.848048	+	0.529919i
0.438371	+	0.898794i
0.961262	+	0.275637i
0.0348995	+	0.999391i
0.961262	−	0.275637i
0.438371	−	0.898794i
0.990268	+	0.139173i
−0.615661	−	0.788011i
−0.882948	+	0.469472i
−0.997564	−	0.0697565i
−0.719340	+	0.694658i
−0.882948	−	0.469472i
−0.241922	+	0.970296i
0.559193	+	0.829038i
−0.615661	+	0.788011i
0.990268	−	0.139173i
0.0348995	−	0.999391i
0.559193	−	0.829038i
−0.241922	−	0.970296i

0.882948

−

0.469472i

−0.615661

0.788011i

−0.0477162

−

0.682374i

0.559193

−

0.829038i

608.1

−0.990268

0.139173i

0.0348995

0.999391i

−0.663721

0.165484i

0.961262

−

0.275637i

614.1

−0.961262

0.275637i

−0.997564

0.0697565i

−0.603541

−

1.13510i

0.848048

−

0.529919i

632.1

0.997564

−

0.0697565i

−0.719340

−

0.694658i

1.55208

1.21262i

0.990268

−

0.139173i

686.1

0.374607

0.927184i

−0.882948

0.469472i

−0.542900

−

1.89332i

−0.719340

0.694658i

977.1

0.997564

0.0697565i

−0.719340

0.694658i

1.55208

−

1.21262i

0.990268

0.139173i

995.1

−0.961262

−

0.275637i

−0.997564

−

0.0697565i

−0.603541

1.13510i

0.848048

0.529919i

1049.1

−0.0348995

−

0.999391i

−0.374607

0.927184i

1.15547

−

0.563559i

−0.997564

0.0697565i

1334.1

−0.848048

−

0.529919i

0.990268

0.139173i

1.63289

−

1.10140i

0.438371

0.898794i

1697.1

0.615661

0.788011i

0.848048

−

0.529919i

0.893036

0.924765i

−0.241922

0.970296i

1703.1

0.719340

0.694658i

0.559193

0.829038i

−0.362486

0.580099i

0.0348995

0.999391i

1721.1

−0.559193

−

0.829038i

−0.241922

0.970296i

−0.178917

1.27306i

−0.374607

0.927184i

1775.1

0.615661

−

0.788011i

0.848048

0.529919i

0.893036

−

0.924765i

−0.241922

−

0.970296i

2066.1

−0.438371

−

0.898794i

0.961262

0.275637i

0.634231

0.256246i

−0.615661

0.788011i

2084.1

0.241922

0.970296i

0.438371

0.898794i

−1.96842

0.0687386i

−0.882948

0.469472i

2138.1

−0.848048

0.529919i

0.990268

−

0.139173i

1.63289

1.10140i

0.438371

−

0.898794i

2423.1

−0.0348995

0.999391i

−0.374607

−

0.927184i

1.15547

0.563559i

−0.997564

−

0.0697565i

2786.1

0.374607

−

0.927184i

−0.882948

−

0.469472i

−0.542900

1.89332i

−0.719340

−

0.694658i

2792.1

0.241922

−

0.970296i

0.438371

−

0.898794i

−1.96842

−

0.0687386i

−0.882948

−

0.469472i

2810.1

−0.438371

0.898794i

0.961262

−

0.275637i

0.634231

−

0.256246i

−0.615661

−

0.788011i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11}) $
11.c	even	5	3	inner
11.d	odd	10	3	inner
27.f	odd	18	1	inner
297.o	even	18	1	inner
297.v	odd	90	3	inner
297.x	even	90	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3267.1.be.a		24
11.b	odd	2	1	CM	3267.1.be.a		24
11.c	even	5	1		3267.1.q.a	✓	6
11.c	even	5	3	inner	3267.1.be.a		24
11.d	odd	10	1		3267.1.q.a	✓	6
11.d	odd	10	3	inner	3267.1.be.a		24
27.f	odd	18	1	inner	3267.1.be.a		24
297.o	even	18	1	inner	3267.1.be.a		24
297.v	odd	90	1		3267.1.q.a	✓	6
297.v	odd	90	3	inner	3267.1.be.a		24
297.x	even	90	1		3267.1.q.a	✓	6
297.x	even	90	3	inner	3267.1.be.a		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3267.1.q.a	✓	6	11.c	even	5	1
3267.1.q.a	✓	6	11.d	odd	10	1
3267.1.q.a	✓	6	297.v	odd	90	1
3267.1.q.a	✓	6	297.x	even	90	1
3267.1.be.a		24	1.a	even	1	1	trivial
3267.1.be.a		24	11.b	odd	2	1	CM
3267.1.be.a		24	11.c	even	5	3	inner
3267.1.be.a		24	11.d	odd	10	3	inner
3267.1.be.a		24	27.f	odd	18	1	inner
3267.1.be.a		24	297.o	even	18	1	inner
3267.1.be.a		24	297.v	odd	90	3	inner
3267.1.be.a		24	297.x	even	90	3	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3267, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{24} $ T^24
$3$	$ T^{24} + T^{21} + \cdots + 1 $ T^24 + T^21 - T^15 - T^12 - T^9 + T^3 + 1
$5$	$ T^{24} - 3 T^{23} + \cdots + 81 $ T^24 - 3T^23 + 3T^22 + 3T^21 - 9T^20 + 54T^19 - 129T^18 + 126T^17 + 63T^16 - 180T^15 + 594T^14 - 1242T^13 + 1629T^12 - 1377T^11 + 1620T^10 - 1026T^9 - 162T^8 + 648T^7 + 297T^6 + 324T^5 - 162T^4 - 162*T^3 + 81
$7$	$ T^{24} $ T^24
$11$	$ T^{24} $ T^24
$13$	$ T^{24} $ T^24
$17$	$ T^{24} $ T^24
$19$	$ T^{24} $ T^24
$23$	$ (T^{6} + 9 T^{3} + 27)^{4} $ (T^6 + 9*T^3 + 27)^4
$29$	$ T^{24} $ T^24
$31$	$ T^{24} + 3 T^{23} + \cdots + 1 $ T^24 + 3T^23 + 3T^22 - T^21 - 9T^20 + 30T^19 + 117T^18 + 126T^17 + 15T^16 - 188T^15 + 330T^14 + 1398T^13 + 1169T^12 - 21T^11 - 996T^10 + 2134T^9 + 6240T^8 + 3816T^7 + 1737T^6 + 672T^5 + 234T^4 + 80T^3 + 24T^2 + 6T + 1
$37$	$ T^{24} - 3 T^{22} + \cdots + 1 $ T^24 - 3T^22 + 2T^21 + 6T^19 + 30T^18 - 45T^17 - 69T^16 + 139T^15 + 288T^14 - 66T^13 - 130T^12 - 639T^11 - 1848T^10 + 4T^9 + 4950T^8 + 1719T^7 + 3T^6 - 198T^5 - 72T^4 - 25T^3 + 3T + 1
$41$	$ T^{24} $ T^24
$43$	$ T^{24} $ T^24
$47$	$ T^{24} + 3 T^{23} + \cdots + 81 $ T^24 + 3T^23 + 3T^22 - 3T^21 - 18T^20 - 9T^19 + 60T^18 + 171T^17 + 225T^16 - 63T^15 - 54T^14 + 594T^13 + 1062T^12 + 1377T^11 + 567T^10 + 297T^9 + 1701T^8 + 2106T^7 + 1998T^6 + 1863T^5 + 1539T^4 + 891T^3 + 486T^2 + 243*T + 81
$53$	$ T^{24} - 6 T^{22} + \cdots + 81 $ T^24 - 6T^22 + 27T^20 - 111T^18 + 441T^16 - 837T^14 + 1386T^12 - 2106T^10 + 2673T^8 - 1242T^6 + 567T^4 - 243*T^2 + 81
$59$	$ T^{24} + 3 T^{23} + \cdots + 81 $ T^24 + 3T^23 + 3T^22 - 3T^21 - 18T^20 - 9T^19 + 60T^18 + 171T^17 + 225T^16 - 63T^15 - 54T^14 + 594T^13 + 1062T^12 + 1377T^11 + 567T^10 + 297T^9 + 1701T^8 + 2106T^7 + 1998T^6 + 1863T^5 + 1539T^4 + 891T^3 + 486T^2 + 243*T + 81
$61$	$ T^{24} $ T^24
$67$	$ (T^{6} + 6 T^{5} + 15 T^{4} + \cdots + 1)^{4} $ (T^6 + 6T^5 + 15T^4 + 19T^3 + 12T^2 + 3*T + 1)^4
$71$	$ T^{24} + 3 T^{22} + \cdots + 81 $ T^24 + 3T^22 - 36T^19 - 30T^18 - 81T^17 - 99T^16 + 81T^15 + 216T^14 + 810T^13 + 900T^12 + 1053T^11 + 1782T^10 + 2430T^9 + 3888T^8 + 3483T^7 + 2403T^6 + 1458T^5 + 810T^4 + 729T^3 + 486T^2 + 243T + 81
$73$	$ T^{24} $ T^24
$79$	$ T^{24} $ T^24
$83$	$ T^{24} $ T^24
$89$	$ (T^{2} + 3 T + 3)^{12} $ (T^2 + 3*T + 3)^12
$97$	$ T^{24} + 6 T^{23} + \cdots + 1 $ T^24 + 6T^23 + 21T^22 + 55T^21 + 117T^20 + 231T^19 + 441T^18 + 801T^17 + 1320T^16 + 1808T^15 + 2301T^14 + 3246T^13 + 5435T^12 + 9102T^11 + 11253T^10 + 9692T^9 + 4512T^8 - 666T^7 - 693T^6 - 87T^5 - 26T^3 - 3T^2 + 3T + 1
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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:	\( N \)	\(=\)	\( 3267 = 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3267.be (of order \(90\), degree \(24\), not minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{90}^{11} q^{3} + \zeta_{90}^{14} q^{4} + (\zeta_{90}^{13} - \zeta_{90}^{3}) q^{5} + \zeta_{90}^{22} q^{9} + \zeta_{90}^{25} q^{12} + (\zeta_{90}^{24} - \zeta_{90}^{14}) q^{15} + \zeta_{90}^{28} q^{16}+ \cdots + ( - \zeta_{90}^{27} + \zeta_{90}^{2}) q^{97}+O(q^{100}) \) q + z^11 * q^3 + z^14 * q^4 + (z^13 - z^3) * q^5 + z^22 * q^9 + z^25 * q^12 + (z^24 - z^14) * q^15 + z^28 * q^16 + (z^27 - z^17) * q^20 + (-z^35 - z^20) * q^23 + (z^26 - z^16 + z^6) * q^25 + z^33 * q^27 + (-z^12 + z^7) * q^31 + z^36 * q^36 + (z^29 + z^19) * q^37 + (z^35 - z^25) * q^45 + (-z^41 + z^21) * q^47 + z^39 * q^48 - z^8 * q^49 + (-z^7 - z^2) * q^53 + (-z^24 - z^19) * q^59 + (z^38 - z^28) * q^60 + z^42 * q^64 + (-z^40 - 1) * q^67 + (-z^31 + z) * q^69 + (-z^43 + z^23) * q^71 + (z^37 - z^27 + z^17) * q^75 + (z^41 - z^31) * q^80 + z^44 * q^81 + (-z^15 - 1) * q^89 + (-z^34 + z^4) * q^92 + (-z^23 + z^18) * q^93 + (-z^27 + z^2) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 3 q^{5} + 3 q^{15} + 6 q^{20} + 3 q^{25} - 3 q^{27} - 3 q^{31} - 6 q^{36} - 3 q^{47} - 3 q^{48} - 3 q^{59} + 3 q^{64} - 24 q^{67} - 6 q^{75} - 36 q^{89} - 6 q^{93} - 6 q^{97}+O(q^{100}) \) 24 * q + 3 * q^5 + 3 * q^15 + 6 * q^20 + 3 * q^25 - 3 * q^27 - 3 * q^31 - 6 * q^36 - 3 * q^47 - 3 * q^48 - 3 * q^59 + 3 * q^64 - 24 * q^67 - 6 * q^75 - 36 * q^89 - 6 * q^93 - 6 * q^97

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	3267.1.be.a

Level	$3267$
Weight	$1$
Character orbit	3267.be
Analytic conductor	$1.630$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{18}$
CM discriminant	-11
Inner twists	$16$

Self dual:	no
Analytic conductor:	\(1.63044539627\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Coefficient field:	\(\Q(\zeta_{45})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 \) x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{18}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

\(n\)	\(244\)	\(3026\)
\(\chi(n)\)	\(-\zeta_{90}^{9}\)	\(\zeta_{90}^{5}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 245.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{24} \) T^24
$3$	\( T^{24} + T^{21} + \cdots + 1 \) T^24 + T^21 - T^15 - T^12 - T^9 + T^3 + 1
$5$	\( T^{24} - 3 T^{23} + \cdots + 81 \) T^24 - 3T^23 + 3T^22 + 3T^21 - 9T^20 + 54T^19 - 129T^18 + 126T^17 + 63T^16 - 180T^15 + 594T^14 - 1242T^13 + 1629T^12 - 1377T^11 + 1620T^10 - 1026T^9 - 162T^8 + 648T^7 + 297T^6 + 324T^5 - 162T^4 - 162*T^3 + 81
$7$	\( T^{24} \) T^24
$11$	\( T^{24} \) T^24
$13$	\( T^{24} \) T^24
$17$	\( T^{24} \) T^24
$19$	\( T^{24} \) T^24
$23$	\( (T^{6} + 9 T^{3} + 27)^{4} \) (T^6 + 9*T^3 + 27)^4
$29$	\( T^{24} \) T^24
$31$	\( T^{24} + 3 T^{23} + \cdots + 1 \) T^24 + 3T^23 + 3T^22 - T^21 - 9T^20 + 30T^19 + 117T^18 + 126T^17 + 15T^16 - 188T^15 + 330T^14 + 1398T^13 + 1169T^12 - 21T^11 - 996T^10 + 2134T^9 + 6240T^8 + 3816T^7 + 1737T^6 + 672T^5 + 234T^4 + 80T^3 + 24T^2 + 6T + 1
$37$	\( T^{24} - 3 T^{22} + \cdots + 1 \) T^24 - 3T^22 + 2T^21 + 6T^19 + 30T^18 - 45T^17 - 69T^16 + 139T^15 + 288T^14 - 66T^13 - 130T^12 - 639T^11 - 1848T^10 + 4T^9 + 4950T^8 + 1719T^7 + 3T^6 - 198T^5 - 72T^4 - 25T^3 + 3T + 1
$41$	\( T^{24} \) T^24
$43$	\( T^{24} \) T^24
$47$	\( T^{24} + 3 T^{23} + \cdots + 81 \) T^24 + 3T^23 + 3T^22 - 3T^21 - 18T^20 - 9T^19 + 60T^18 + 171T^17 + 225T^16 - 63T^15 - 54T^14 + 594T^13 + 1062T^12 + 1377T^11 + 567T^10 + 297T^9 + 1701T^8 + 2106T^7 + 1998T^6 + 1863T^5 + 1539T^4 + 891T^3 + 486T^2 + 243*T + 81
$53$	\( T^{24} - 6 T^{22} + \cdots + 81 \) T^24 - 6T^22 + 27T^20 - 111T^18 + 441T^16 - 837T^14 + 1386T^12 - 2106T^10 + 2673T^8 - 1242T^6 + 567T^4 - 243*T^2 + 81
$59$	\( T^{24} + 3 T^{23} + \cdots + 81 \) T^24 + 3T^23 + 3T^22 - 3T^21 - 18T^20 - 9T^19 + 60T^18 + 171T^17 + 225T^16 - 63T^15 - 54T^14 + 594T^13 + 1062T^12 + 1377T^11 + 567T^10 + 297T^9 + 1701T^8 + 2106T^7 + 1998T^6 + 1863T^5 + 1539T^4 + 891T^3 + 486T^2 + 243*T + 81
$61$	\( T^{24} \) T^24
$67$	\( (T^{6} + 6 T^{5} + 15 T^{4} + \cdots + 1)^{4} \) (T^6 + 6T^5 + 15T^4 + 19T^3 + 12T^2 + 3*T + 1)^4
$71$	\( T^{24} + 3 T^{22} + \cdots + 81 \) T^24 + 3T^22 - 36T^19 - 30T^18 - 81T^17 - 99T^16 + 81T^15 + 216T^14 + 810T^13 + 900T^12 + 1053T^11 + 1782T^10 + 2430T^9 + 3888T^8 + 3483T^7 + 2403T^6 + 1458T^5 + 810T^4 + 729T^3 + 486T^2 + 243T + 81
$73$	\( T^{24} \) T^24
$79$	\( T^{24} \) T^24
$83$	\( T^{24} \) T^24
$89$	\( (T^{2} + 3 T + 3)^{12} \) (T^2 + 3*T + 3)^12
$97$	\( T^{24} + 6 T^{23} + \cdots + 1 \) T^24 + 6T^23 + 21T^22 + 55T^21 + 117T^20 + 231T^19 + 441T^18 + 801T^17 + 1320T^16 + 1808T^15 + 2301T^14 + 3246T^13 + 5435T^12 + 9102T^11 + 11253T^10 + 9692T^9 + 4512T^8 - 666T^7 - 693T^6 - 87T^5 - 26T^3 - 3T^2 + 3T + 1
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