LMFDB - Newform orbit 344.1.be.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [344,1,Mod(67,344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("344.67"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(344, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([21, 21, 40])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 344 = 2^{3} \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	344.be (of order $42$, degree $12$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.171678364346$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{21}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{21} - \cdots)$

$q$-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_{42}^{15} q^{2} + ( - \zeta_{42}^{11} + \zeta_{42}^{8}) q^{3} - \zeta_{42}^{9} q^{4} + ( - \zeta_{42}^{5} + \zeta_{42}^{2}) q^{6} - \zeta_{42}^{3} q^{8} + ( - \zeta_{42}^{19} + \cdots - \zeta_{42}) q^{9} + \cdots + (\zeta_{42}^{20} + \zeta_{42}^{18} + \cdots + 1) q^{99} +O(q^{100}) $ q - z^15 * q^2 + (-z^11 + z^8) * q^3 - z^9 * q^4 + (-z^5 + z^2) * q^6 - z^3 * q^8 + (-z^19 + z^16 - z) * q^9 + (z^20 + z^4) * q^11 + (z^20 - z^17) * q^12 + z^18 * q^16 + (z^6 + z^2) * q^17 + (z^16 - z^13 + z^10) * q^18 + (z^18 - z^7) * q^19 + (-z^19 + z^14) * q^22 + (z^14 - z^11) * q^24 - z^13 * q^25 + (z^12 - z^9 + z^6 - z^3) * q^27 + z^12 * q^32 + (-z^15 + z^12 + z^10 - z^7) * q^33 + (-z^17 + 1) * q^34 + (z^10 - z^7 + z^4) * q^36 + (z^12 - z) * q^38 + (-z^5 + z^4) * q^41 - z * q^43 + (-z^13 + z^8) * q^44 + (z^8 - z^5) * q^48 + z^14 * q^49 - z^7 * q^50 + (-z^17 + z^14 - z^13 + z^10) * q^51 + (z^18 + z^6 - z^3 + 1) * q^54 + (z^18 - z^15 + z^8 - z^5) * q^57 + (-z^19 - z^17) * q^59 + z^6 * q^64 + (-z^9 + z^6 + z^4 - z) * q^66 - z^2 * q^67 + (-z^15 - z^11) * q^68 + (-z^19 + z^4 - z) * q^72 - z^10 * q^73 + (-z^3 + 1) * q^75 + (z^16 + z^6) * q^76 + (z^20 - z^17 + z^14 - z^11 + z^2) * q^81 + (z^20 - z^19) * q^82 + (z^14 - z^5) * q^83 + z^16 * q^86 + (-z^7 + z^2) * q^88 + (z^10 - z^9) * q^89 + (z^20 + z^2) * q^96 + (-z^13 - z^11) * q^97 + z^8 * q^98 + (z^20 + z^18 - z^15 - z^5 + z^2 + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} - 2 q^{8} + 3 q^{9} + 2 q^{11} + 2 q^{12} - 2 q^{16} - q^{17} + 3 q^{18} - 8 q^{19} - 5 q^{22} - 5 q^{24} + q^{25} - 8 q^{27} - 2 q^{32} - 9 q^{33} + 13 q^{34}+ \cdots + 11 q^{99}+O(q^{100}) $ 12 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^6 - 2 * q^8 + 3 * q^9 + 2 * q^11 + 2 * q^12 - 2 * q^16 - q^17 + 3 * q^18 - 8 * q^19 - 5 * q^22 - 5 * q^24 + q^25 - 8 * q^27 - 2 * q^32 - 9 * q^33 + 13 * q^34 - 4 * q^36 - q^38 + 2 * q^41 + q^43 + 2 * q^44 + 2 * q^48 - 6 * q^49 - 6 * q^50 - 3 * q^51 + 6 * q^54 - 2 * q^57 + 2 * q^59 - 2 * q^64 - 2 * q^66 - q^67 - q^68 + 3 * q^72 - q^73 + 10 * q^75 - q^76 - 2 * q^81 + 2 * q^82 - 5 * q^83 + q^86 - 5 * q^88 - q^89 + 2 * q^96 + 2 * q^97 + q^98 + 11 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/344\mathbb{Z}\right)^\times$.

$n$	$87$	$89$	$173$
$\chi(n)$	$-1$	$-\zeta_{42}^{19}$	$-1$

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} $

67.1

−0.988831	−	0.149042i
0.0747301	+	0.997204i
−0.733052	−	0.680173i
−0.733052	+	0.680173i
0.365341	+	0.930874i
0.365341	−	0.930874i
0.826239	+	0.563320i
−0.988831	+	0.149042i
0.955573	+	0.294755i
0.826239	−	0.563320i
0.0747301	−	0.997204i
0.955573	−	0.294755i

0.623490

−

0.781831i

0.440071

−

0.0663300i

−0.222521

−

0.974928i

0.222521

−

0.385418i

−0.900969

−

0.433884i

−0.766310

0.236375i

83.1

−0.900969

−

0.433884i

0.0931869

−

1.24349i

0.623490

0.781831i

−0.623490

1.07992i

−0.222521

−

0.974928i

−0.548760

−

0.0827122i

99.1

−0.222521

0.974928i

1.32091

−

1.22563i

−0.900969

−

0.433884i

0.900969

1.56052i

0.623490

−

0.781831i

0.167917

−

2.24070i

139.1

−0.222521

−

0.974928i

1.32091

1.22563i

−0.900969

0.433884i

0.900969

−

1.56052i

0.623490

0.781831i

0.167917

2.24070i

187.1

0.623490

−

0.781831i

−0.162592

0.414278i

−0.222521

−

0.974928i

0.222521

0.385418i

−0.900969

−

0.433884i

0.587862

0.545456i

195.1

0.623490

0.781831i

−0.162592

−

0.414278i

−0.222521

0.974928i

0.222521

−

0.385418i

−0.900969

0.433884i

0.587862

−

0.545456i

203.1

−0.900969

0.433884i

1.03030

−

0.702449i

0.623490

−

0.781831i

−0.623490

1.07992i

−0.222521

0.974928i

0.202749

−

0.516596i

267.1

0.623490

0.781831i

0.440071

0.0663300i

−0.222521

0.974928i

0.222521

0.385418i

−0.900969

0.433884i

−0.766310

−

0.236375i

275.1

−0.222521

−

0.974928i

−1.72188

0.531130i

−0.900969

0.433884i

0.900969

1.56052i

0.623490

0.781831i

1.85654

−

1.26577i

283.1

−0.900969

−

0.433884i

1.03030

0.702449i

0.623490

0.781831i

−0.623490

−

1.07992i

−0.222521

−

0.974928i

0.202749

0.516596i

315.1

−0.900969

0.433884i

0.0931869

1.24349i

0.623490

−

0.781831i

−0.623490

−

1.07992i

−0.222521

0.974928i

−0.548760

0.0827122i

339.1

−0.222521

0.974928i

−1.72188

−

0.531130i

−0.900969

−

0.433884i

0.900969

−

1.56052i

0.623490

−

0.781831i

1.85654

1.26577i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
43.g	even	21	1	inner
344.be	odd	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	344.1.be.a	✓	12
3.b	odd	2	1		3096.1.eq.a		12
4.b	odd	2	1		1376.1.by.a		12
8.b	even	2	1		1376.1.by.a		12
8.d	odd	2	1	CM	344.1.be.a	✓	12
24.f	even	2	1		3096.1.eq.a		12
43.g	even	21	1	inner	344.1.be.a	✓	12
129.o	odd	42	1		3096.1.eq.a		12
172.o	odd	42	1		1376.1.by.a		12
344.z	even	42	1		1376.1.by.a		12
344.be	odd	42	1	inner	344.1.be.a	✓	12
1032.ci	even	42	1		3096.1.eq.a		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
344.1.be.a	✓	12	1.a	even	1	1	trivial
344.1.be.a	✓	12	8.d	odd	2	1	CM
344.1.be.a	✓	12	43.g	even	21	1	inner
344.1.be.a	✓	12	344.be	odd	42	1	inner
1376.1.by.a		12	4.b	odd	2	1
1376.1.by.a		12	8.b	even	2	1
1376.1.by.a		12	172.o	odd	42	1
1376.1.by.a		12	344.z	even	42	1
3096.1.eq.a		12	3.b	odd	2	1
3096.1.eq.a		12	24.f	even	2	1
3096.1.eq.a		12	129.o	odd	42	1
3096.1.eq.a		12	1032.ci	even	42	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(344, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} $ (T^6 + T^5 + T^4 + T^3 + T^2 + T + 1)^2
$3$	$ T^{12} - 2 T^{11} + \cdots + 1 $ T^12 - 2T^11 + 8T^9 - 9T^8 + 22T^6 - 42T^5 + 45T^4 - 20T^3 + 7T^2 - 4*T + 1
$5$	$ T^{12} $ T^12
$7$	$ T^{12} $ T^12
$11$	$ T^{12} - 2 T^{11} + \cdots + 1 $ T^12 - 2T^11 + 3T^10 + 3T^9 - 9T^8 + T^7 + 49T^6 - 97T^5 + 96T^4 - 46T^3 + 52T^2 - 9T + 1
$13$	$ T^{12} $ T^12
$17$	$ T^{12} + T^{11} + \cdots + 1 $ T^12 + T^11 + 6T^9 + 6T^8 + 7T^7 + 43T^6 + 35T^5 + 90T^4 + 104T^3 + 70T^2 + 15*T + 1
$19$	$ T^{12} + 8 T^{11} + \cdots + 1 $ T^12 + 8T^11 + 35T^10 + 104T^9 + 230T^8 + 392T^7 + 519T^6 + 518T^5 + 349T^4 + 118T^3 - 6T + 1
$23$	$ T^{12} $ T^12
$29$	$ T^{12} $ T^12
$31$	$ T^{12} $ T^12
$37$	$ T^{12} $ T^12
$41$	$ T^{12} - 2 T^{11} + \cdots + 1 $ T^12 - 2T^11 + 3T^10 + 3T^9 - 9T^8 + T^7 + 49T^6 - 97T^5 + 96T^4 - 46T^3 + 52T^2 - 9T + 1
$43$	$ T^{12} - T^{11} + \cdots + 1 $ T^12 - T^11 + T^9 - T^8 + T^6 - T^4 + T^3 - T + 1
$47$	$ T^{12} $ T^12
$53$	$ T^{12} $ T^12
$59$	$ T^{12} - 2 T^{11} + \cdots + 1 $ T^12 - 2T^11 + 3T^10 - 4T^9 + 12T^8 - 6T^7 + 7T^6 - 6T^5 + 54T^4 + 94T^3 + 52T^2 + 5*T + 1
$61$	$ T^{12} $ T^12
$67$	$ T^{12} + T^{11} + \cdots + 1 $ T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1
$71$	$ T^{12} $ T^12
$73$	$ T^{12} + T^{11} + \cdots + 1 $ T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1
$79$	$ T^{12} $ T^12
$83$	$ T^{12} + 5 T^{11} + \cdots + 1 $ T^12 + 5T^11 + 14T^10 + 29T^9 + 47T^8 + 56T^7 + 57T^6 + 56T^5 + 31T^4 + T^3 + 3*T + 1
$89$	$ T^{12} + T^{11} + \cdots + 1 $ T^12 + T^11 + 6T^9 + 6T^8 + 7T^7 + 43T^6 + 35T^5 + 90T^4 + 104T^3 + 70T^2 + 15*T + 1
$97$	$ T^{12} - 2 T^{11} + \cdots + 1 $ T^12 - 2T^11 + 3T^10 + 3T^9 - 9T^8 + T^7 + 49T^6 - 97T^5 + 96T^4 - 46T^3 + 52T^2 - 9T + 1
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Level:	\( N \)	\(=\)	\( 344 = 2^{3} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	344.be (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{42}^{15} q^{2} + ( - \zeta_{42}^{11} + \zeta_{42}^{8}) q^{3} - \zeta_{42}^{9} q^{4} + ( - \zeta_{42}^{5} + \zeta_{42}^{2}) q^{6} - \zeta_{42}^{3} q^{8} + ( - \zeta_{42}^{19} + \cdots - \zeta_{42}) q^{9} + \cdots + (\zeta_{42}^{20} + \zeta_{42}^{18} + \cdots + 1) q^{99} +O(q^{100}) \) q - z^15 * q^2 + (-z^11 + z^8) * q^3 - z^9 * q^4 + (-z^5 + z^2) * q^6 - z^3 * q^8 + (-z^19 + z^16 - z) * q^9 + (z^20 + z^4) * q^11 + (z^20 - z^17) * q^12 + z^18 * q^16 + (z^6 + z^2) * q^17 + (z^16 - z^13 + z^10) * q^18 + (z^18 - z^7) * q^19 + (-z^19 + z^14) * q^22 + (z^14 - z^11) * q^24 - z^13 * q^25 + (z^12 - z^9 + z^6 - z^3) * q^27 + z^12 * q^32 + (-z^15 + z^12 + z^10 - z^7) * q^33 + (-z^17 + 1) * q^34 + (z^10 - z^7 + z^4) * q^36 + (z^12 - z) * q^38 + (-z^5 + z^4) * q^41 - z * q^43 + (-z^13 + z^8) * q^44 + (z^8 - z^5) * q^48 + z^14 * q^49 - z^7 * q^50 + (-z^17 + z^14 - z^13 + z^10) * q^51 + (z^18 + z^6 - z^3 + 1) * q^54 + (z^18 - z^15 + z^8 - z^5) * q^57 + (-z^19 - z^17) * q^59 + z^6 * q^64 + (-z^9 + z^6 + z^4 - z) * q^66 - z^2 * q^67 + (-z^15 - z^11) * q^68 + (-z^19 + z^4 - z) * q^72 - z^10 * q^73 + (-z^3 + 1) * q^75 + (z^16 + z^6) * q^76 + (z^20 - z^17 + z^14 - z^11 + z^2) * q^81 + (z^20 - z^19) * q^82 + (z^14 - z^5) * q^83 + z^16 * q^86 + (-z^7 + z^2) * q^88 + (z^10 - z^9) * q^89 + (z^20 + z^2) * q^96 + (-z^13 - z^11) * q^97 + z^8 * q^98 + (z^20 + z^18 - z^15 - z^5 + z^2 + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} - 2 q^{8} + 3 q^{9} + 2 q^{11} + 2 q^{12} - 2 q^{16} - q^{17} + 3 q^{18} - 8 q^{19} - 5 q^{22} - 5 q^{24} + q^{25} - 8 q^{27} - 2 q^{32} - 9 q^{33} + 13 q^{34}+ \cdots + 11 q^{99}+O(q^{100}) \) 12 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^6 - 2 * q^8 + 3 * q^9 + 2 * q^11 + 2 * q^12 - 2 * q^16 - q^17 + 3 * q^18 - 8 * q^19 - 5 * q^22 - 5 * q^24 + q^25 - 8 * q^27 - 2 * q^32 - 9 * q^33 + 13 * q^34 - 4 * q^36 - q^38 + 2 * q^41 + q^43 + 2 * q^44 + 2 * q^48 - 6 * q^49 - 6 * q^50 - 3 * q^51 + 6 * q^54 - 2 * q^57 + 2 * q^59 - 2 * q^64 - 2 * q^66 - q^67 - q^68 + 3 * q^72 - q^73 + 10 * q^75 - q^76 - 2 * q^81 + 2 * q^82 - 5 * q^83 + q^86 - 5 * q^88 - q^89 + 2 * q^96 + 2 * q^97 + q^98 + 11 * q^99

Introduction

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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	344.1.be.a

Level	$344$
Weight	$1$
Character orbit	344.be
Analytic conductor	$0.172$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{21}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.171678364346\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\Q(\zeta_{21})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{21}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{21} - \cdots)\)

\(n\)	\(87\)	\(89\)	\(173\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{42}^{19}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 67.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
43.g	even	21	1	inner
344.be	odd	42	1	inner

$p$	$F_p(T)$
$2$	\( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) (T^6 + T^5 + T^4 + T^3 + T^2 + T + 1)^2
$3$	\( T^{12} - 2 T^{11} + \cdots + 1 \) T^12 - 2T^11 + 8T^9 - 9T^8 + 22T^6 - 42T^5 + 45T^4 - 20T^3 + 7T^2 - 4*T + 1
$5$	\( T^{12} \) T^12
$7$	\( T^{12} \) T^12
$11$	\( T^{12} - 2 T^{11} + \cdots + 1 \) T^12 - 2T^11 + 3T^10 + 3T^9 - 9T^8 + T^7 + 49T^6 - 97T^5 + 96T^4 - 46T^3 + 52T^2 - 9T + 1
$13$	\( T^{12} \) T^12
$17$	\( T^{12} + T^{11} + \cdots + 1 \) T^12 + T^11 + 6T^9 + 6T^8 + 7T^7 + 43T^6 + 35T^5 + 90T^4 + 104T^3 + 70T^2 + 15*T + 1
$19$	\( T^{12} + 8 T^{11} + \cdots + 1 \) T^12 + 8T^11 + 35T^10 + 104T^9 + 230T^8 + 392T^7 + 519T^6 + 518T^5 + 349T^4 + 118T^3 - 6T + 1
$23$	\( T^{12} \) T^12
$29$	\( T^{12} \) T^12
$31$	\( T^{12} \) T^12
$37$	\( T^{12} \) T^12
$41$	\( T^{12} - 2 T^{11} + \cdots + 1 \) T^12 - 2T^11 + 3T^10 + 3T^9 - 9T^8 + T^7 + 49T^6 - 97T^5 + 96T^4 - 46T^3 + 52T^2 - 9T + 1
$43$	\( T^{12} - T^{11} + \cdots + 1 \) T^12 - T^11 + T^9 - T^8 + T^6 - T^4 + T^3 - T + 1
$47$	\( T^{12} \) T^12
$53$	\( T^{12} \) T^12
$59$	\( T^{12} - 2 T^{11} + \cdots + 1 \) T^12 - 2T^11 + 3T^10 - 4T^9 + 12T^8 - 6T^7 + 7T^6 - 6T^5 + 54T^4 + 94T^3 + 52T^2 + 5*T + 1
$61$	\( T^{12} \) T^12
$67$	\( T^{12} + T^{11} + \cdots + 1 \) T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1
$71$	\( T^{12} \) T^12
$73$	\( T^{12} + T^{11} + \cdots + 1 \) T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1
$79$	\( T^{12} \) T^12
$83$	\( T^{12} + 5 T^{11} + \cdots + 1 \) T^12 + 5T^11 + 14T^10 + 29T^9 + 47T^8 + 56T^7 + 57T^6 + 56T^5 + 31T^4 + T^3 + 3*T + 1
$89$	\( T^{12} + T^{11} + \cdots + 1 \) T^12 + T^11 + 6T^9 + 6T^8 + 7T^7 + 43T^6 + 35T^5 + 90T^4 + 104T^3 + 70T^2 + 15*T + 1
$97$	\( T^{12} - 2 T^{11} + \cdots + 1 \) T^12 - 2T^11 + 3T^10 + 3T^9 - 9T^8 + T^7 + 49T^6 - 97T^5 + 96T^4 - 46T^3 + 52T^2 - 9T + 1
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