LMFDB - Newform orbit 675.2.f.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(107,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("675.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.f (of order $4$, degree $2$, 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:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.3317760000.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 2x^{6} + 8x^{5} + 13x^{4} - 44x^{3} + 164x^{2} - 140x + 145 $ x^8 - 4x^7 + 2x^6 + 8x^5 + 13x^4 - 44x^3 + 164x^2 - 140*x + 145
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{4} q^{2} + (\beta_{2} + 3 \beta_1) q^{4} + ( - \beta_{5} - 3 \beta_{3}) q^{8} + ( - 3 \beta_{6} - 7) q^{16} + (\beta_{7} + \beta_{4}) q^{17} + (2 \beta_{2} - \beta_1) q^{19} + ( - \beta_{5} + 3 \beta_{3}) q^{23}+ \cdots - 7 \beta_{3} q^{98}+O(q^{100}) $ q + b4 * q^2 + (b2 + 3b1) q^4 + (-b5 - 3b3) q^8 + (-3b6 - 7) q^16 + (b7 + b4) * q^17 + (2b2 - b1) q^19 + (-b5 + 3b3) q^23 + (2b6 + 3) q^31 + (b7 - 10b4) q^32 + (-b2 + 4b1) q^34 + (-2b5 - 3b3) * q^38 + (b6 + 13) * q^46 + (2b7 - 2b4) * q^47 + 7b1 q^49 + (b5 + 3b3) q^53 + (4b6 - 3) q^61 + (-2b7 + 9b4) * q^62 + (-6b2 - 31b1) * q^64 - b5 * q^68 + (-3b6 - 16) q^76 + (-2b2 + 7b1) * q^79 + (b5 - 7b3) q^83 + (-3b7 + 10b4) * q^92 + (-6b2 - 12b1) * q^94 - 7b3 q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 68 q^{16} + 32 q^{31} + 108 q^{46} - 8 q^{61} - 140 q^{76}+O(q^{100}) $ 8 * q - 68 * q^16 + 32 * q^31 + 108 * q^46 - 8 * q^61 - 140 * q^76

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 4x^{7} + 2x^{6} + 8x^{5} + 13x^{4} - 44x^{3} + 164x^{2} - 140x + 145 $ x^8 - 4*x^7 + 2*x^6 + 8*x^5 + 13*x^4 - 44*x^3 + 164*x^2 - 140*x + 145 :

$\beta_{1}$	$=$	$ ( -2\nu^{6} + 6\nu^{5} + 5\nu^{4} - 20\nu^{3} - 59\nu^{2} + 70\nu - 164 ) / 33 $ (-2v^6 + 6v^5 + 5v^4 - 20v^3 - 59v^2 + 70v - 164) / 33
$\beta_{2}$	$=$	$ ( 132\nu^{7} - 293\nu^{6} - 474\nu^{5} + 650\nu^{4} + 2548\nu^{3} + 2395\nu^{2} + 15832\nu + 3463 ) / 5577 $ (132v^7 - 293v^6 - 474v^5 + 650v^4 + 2548v^3 + 2395v^2 + 15832*v + 3463) / 5577
$\beta_{3}$	$=$	$ ( -50\nu^{7} + 6\nu^{6} + 579\nu^{5} - 195\nu^{4} - 2860\nu^{3} - 1596\nu^{2} + 973\nu - 13216 ) / 1859 $ (-50v^7 + 6v^6 + 579v^5 - 195v^4 - 2860v^3 - 1596v^2 + 973*v - 13216) / 1859
$\beta_{4}$	$=$	$ ( -191\nu^{7} + 1091\nu^{6} - 1104\nu^{5} - 3601\nu^{4} + 871\nu^{3} + 21768\nu^{2} - 36157\nu + 38659 ) / 5577 $ (-191v^7 + 1091v^6 - 1104v^5 - 3601v^4 + 871v^3 + 21768v^2 - 36157*v + 38659) / 5577
$\beta_{5}$	$=$	$ ( -250\nu^{7} + 1044\nu^{6} - 147\nu^{5} - 3510\nu^{4} - 4160\nu^{3} + 16356\nu^{2} - 25048\nu + 22645 ) / 1859 $ (-250v^7 + 1044v^6 - 147v^5 - 3510v^4 - 4160v^3 + 16356v^2 - 25048*v + 22645) / 1859
$\beta_{6}$	$=$	$ ( 300\nu^{7} - 1050\nu^{6} - 432\nu^{5} + 3705\nu^{4} + 7020\nu^{3} - 14760\nu^{2} + 29652\nu - 11288 ) / 1859 $ (300v^7 - 1050v^6 - 432v^5 + 3705v^4 + 7020v^3 - 14760v^2 + 29652*v - 11288) / 1859
$\beta_{7}$	$=$	$ ( 955\nu^{7} - 2920\nu^{6} - 2085\nu^{5} + 8879\nu^{4} + 26572\nu^{3} - 31269\nu^{2} + 86483\nu - 13310 ) / 5577 $ (955v^7 - 2920v^6 - 2085v^5 + 8879v^4 + 26572v^3 - 31269v^2 + 86483*v - 13310) / 5577

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{6} + \beta_{5} + \beta_{3} + 1 ) / 3 $ (b6 + b5 + b3 + 1) / 3
$\nu^{2}$	$=$	$ ( \beta_{6} + 6\beta_{3} - 9\beta _1 + 4 ) / 3 $ (b6 + 6b3 - 9b1 + 4) / 3
$\nu^{3}$	$=$	$ ( 3\beta_{7} + 2\beta_{6} + 3\beta_{5} - 3\beta_{4} + 12\beta_{3} - 9\beta_{2} - 18\beta _1 + 5 ) / 3 $ (3b7 + 2b6 + 3b5 - 3b4 + 12b3 - 9b2 - 18*b1 + 5) / 3
$\nu^{4}$	$=$	$ ( 3\beta_{6} + 4\beta_{5} - 36\beta_{4} + 28\beta_{3} - 18\beta_{2} - 90\beta _1 - 18 ) / 3 $ (3b6 + 4b5 - 36b4 + 28b3 - 18b2 - 90b1 - 18) / 3
$\nu^{5}$	$=$	$ ( 30\beta_{7} - 40\beta_{6} + \beta_{5} - 120\beta_{4} + 46\beta_{3} - 60\beta_{2} - 210\beta _1 - 31 ) / 3 $ (30b7 - 40b6 + b5 - 120b4 + 46b3 - 60b2 - 210b1 - 31) / 3
$\nu^{6}$	$=$	$ ( 60\beta_{7} - 127\beta_{6} + 18\beta_{5} - 420\beta_{4} - 54\beta_{3} - 135\beta_{2} - 459\beta _1 - 517 ) / 3 $ (60b7 - 127b6 + 18b5 - 420b4 - 54b3 - 135b2 - 459*b1 - 517) / 3
$\nu^{7}$	$=$	$ ( 183\beta_{7} - 617\beta_{6} - 154\beta_{5} - 1128\beta_{4} - 553\beta_{3} - 126\beta_{2} - 819\beta _1 - 1538 ) / 3 $ (183b7 - 617b6 - 154b5 - 1128b4 - 553b3 - 126b2 - 819*b1 - 1538) / 3

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$-1$	$-\beta_{1}$

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} $

107.1

0.393289	−	1.22474i
0.606711	+	1.22474i
−1.84278	−	1.22474i
2.84278	+	1.22474i
0.393289	+	1.22474i
0.606711	−	1.22474i
−1.84278	+	1.22474i
2.84278	−	1.22474i

−1.98168

1.98168i

−

5.85410i

7.63759

7.63759i

107.2

−0.756934

0.756934i

0.854102i

−2.16037

−

2.16037i

107.3

0.756934

−

0.756934i

0.854102i

2.16037

2.16037i

107.4

1.98168

−

1.98168i

−

5.85410i

−7.63759

−

7.63759i

593.1

−1.98168

−

1.98168i

5.85410i

7.63759

−

7.63759i

593.2

−0.756934

−

0.756934i

−

0.854102i

−2.16037

2.16037i

593.3

0.756934

0.756934i

−

0.854102i

2.16037

−

2.16037i

593.4

1.98168

1.98168i

5.85410i

−7.63759

7.63759i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
3.b	odd	2	1	inner
5.b	even	2	1	inner
5.c	odd	4	2	inner
15.e	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	675.2.f.h	✓	8
3.b	odd	2	1	inner	675.2.f.h	✓	8
5.b	even	2	1	inner	675.2.f.h	✓	8
5.c	odd	4	2	inner	675.2.f.h	✓	8
15.d	odd	2	1	CM	675.2.f.h	✓	8
15.e	even	4	2	inner	675.2.f.h	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
675.2.f.h	✓	8	1.a	even	1	1	trivial
675.2.f.h	✓	8	3.b	odd	2	1	inner
675.2.f.h	✓	8	5.b	even	2	1	inner
675.2.f.h	✓	8	5.c	odd	4	2	inner
675.2.f.h	✓	8	15.d	odd	2	1	CM
675.2.f.h	✓	8	15.e	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(675, [\chi])$:

$ T_{2}^{8} + 63T_{2}^{4} + 81 $

$ T_{7} $

$ T_{29} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 63T^{4} + 81 $ T^8 + 63*T^4 + 81
$3$	$ T^{8} $ T^8
$5$	$ T^{8} $ T^8
$7$	$ T^{8} $ T^8
$11$	$ T^{8} $ T^8
$13$	$ T^{8} $ T^8
$17$	$ T^{8} + 2898T^{4} + 81 $ T^8 + 2898*T^4 + 81
$19$	$ (T^{4} + 98 T^{2} + 1681)^{2} $ (T^4 + 98*T^2 + 1681)^2
$23$	$ T^{8} + 9378 T^{4} + 10556001 $ T^8 + 9378*T^4 + 10556001
$29$	$ T^{8} $ T^8
$31$	$ (T^{2} - 8 T - 29)^{4} $ (T^2 - 8*T - 29)^4
$37$	$ T^{8} $ T^8
$41$	$ T^{8} $ T^8
$43$	$ T^{8} $ T^8
$47$	$ (T^{4} + 11664)^{2} $ (T^4 + 11664)^2
$53$	$ T^{8} + 13698 T^{4} + 1185921 $ T^8 + 13698*T^4 + 1185921
$59$	$ T^{8} $ T^8
$61$	$ (T^{2} + 2 T - 179)^{4} $ (T^2 + 2*T - 179)^4
$67$	$ T^{8} $ T^8
$71$	$ T^{8} $ T^8
$73$	$ T^{8} $ T^8
$79$	$ (T^{4} + 218 T^{2} + 361)^{2} $ (T^4 + 218*T^2 + 361)^2
$83$	$ T^{8} + \cdots + 3154956561 $ T^8 + 123858*T^4 + 3154956561
$89$	$ T^{8} $ T^8
$97$	$ T^{8} $ T^8
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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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{4} q^{2} + (\beta_{2} + 3 \beta_1) q^{4} + ( - \beta_{5} - 3 \beta_{3}) q^{8} + ( - 3 \beta_{6} - 7) q^{16} + (\beta_{7} + \beta_{4}) q^{17} + (2 \beta_{2} - \beta_1) q^{19} + ( - \beta_{5} + 3 \beta_{3}) q^{23}+ \cdots - 7 \beta_{3} q^{98}+O(q^{100}) \) q + b4 * q^2 + (b2 + 3b1) q^4 + (-b5 - 3b3) q^8 + (-3b6 - 7) q^16 + (b7 + b4) * q^17 + (2b2 - b1) q^19 + (-b5 + 3b3) q^23 + (2b6 + 3) q^31 + (b7 - 10b4) q^32 + (-b2 + 4b1) q^34 + (-2b5 - 3b3) * q^38 + (b6 + 13) * q^46 + (2b7 - 2b4) * q^47 + 7b1 q^49 + (b5 + 3b3) q^53 + (4b6 - 3) q^61 + (-2b7 + 9b4) * q^62 + (-6b2 - 31b1) * q^64 - b5 * q^68 + (-3b6 - 16) q^76 + (-2b2 + 7b1) * q^79 + (b5 - 7b3) q^83 + (-3b7 + 10b4) * q^92 + (-6b2 - 12b1) * q^94 - 7b3 q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 68 q^{16} + 32 q^{31} + 108 q^{46} - 8 q^{61} - 140 q^{76}+O(q^{100}) \) 8 * q - 68 * q^16 + 32 * q^31 + 108 * q^46 - 8 * q^61 - 140 * q^76

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	675.2.f.h

Level	$675$
Weight	$2$
Character orbit	675.f
Analytic conductor	$5.390$
Analytic rank	$0$
Dimension	$8$
CM discriminant	-15
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(5.38990213644\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(i)\)
Coefficient field:	8.0.3317760000.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - 4x^{7} + 2x^{6} + 8x^{5} + 13x^{4} - 44x^{3} + 164x^{2} - 140x + 145 \) x^8 - 4x^7 + 2x^6 + 8x^5 + 13x^4 - 44x^3 + 164x^2 - 140*x + 145
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

\(\beta_{1}\)	\(=\)	\( ( -2\nu^{6} + 6\nu^{5} + 5\nu^{4} - 20\nu^{3} - 59\nu^{2} + 70\nu - 164 ) / 33 \) (-2v^6 + 6v^5 + 5v^4 - 20v^3 - 59v^2 + 70v - 164) / 33
\(\beta_{2}\)	\(=\)	\( ( 132\nu^{7} - 293\nu^{6} - 474\nu^{5} + 650\nu^{4} + 2548\nu^{3} + 2395\nu^{2} + 15832\nu + 3463 ) / 5577 \) (132v^7 - 293v^6 - 474v^5 + 650v^4 + 2548v^3 + 2395v^2 + 15832*v + 3463) / 5577
\(\beta_{3}\)	\(=\)	\( ( -50\nu^{7} + 6\nu^{6} + 579\nu^{5} - 195\nu^{4} - 2860\nu^{3} - 1596\nu^{2} + 973\nu - 13216 ) / 1859 \) (-50v^7 + 6v^6 + 579v^5 - 195v^4 - 2860v^3 - 1596v^2 + 973*v - 13216) / 1859
\(\beta_{4}\)	\(=\)	\( ( -191\nu^{7} + 1091\nu^{6} - 1104\nu^{5} - 3601\nu^{4} + 871\nu^{3} + 21768\nu^{2} - 36157\nu + 38659 ) / 5577 \) (-191v^7 + 1091v^6 - 1104v^5 - 3601v^4 + 871v^3 + 21768v^2 - 36157*v + 38659) / 5577
\(\beta_{5}\)	\(=\)	\( ( -250\nu^{7} + 1044\nu^{6} - 147\nu^{5} - 3510\nu^{4} - 4160\nu^{3} + 16356\nu^{2} - 25048\nu + 22645 ) / 1859 \) (-250v^7 + 1044v^6 - 147v^5 - 3510v^4 - 4160v^3 + 16356v^2 - 25048*v + 22645) / 1859
\(\beta_{6}\)	\(=\)	\( ( 300\nu^{7} - 1050\nu^{6} - 432\nu^{5} + 3705\nu^{4} + 7020\nu^{3} - 14760\nu^{2} + 29652\nu - 11288 ) / 1859 \) (300v^7 - 1050v^6 - 432v^5 + 3705v^4 + 7020v^3 - 14760v^2 + 29652*v - 11288) / 1859
\(\beta_{7}\)	\(=\)	\( ( 955\nu^{7} - 2920\nu^{6} - 2085\nu^{5} + 8879\nu^{4} + 26572\nu^{3} - 31269\nu^{2} + 86483\nu - 13310 ) / 5577 \) (955v^7 - 2920v^6 - 2085v^5 + 8879v^4 + 26572v^3 - 31269v^2 + 86483*v - 13310) / 5577

\(\nu\)	\(=\)	\( ( \beta_{6} + \beta_{5} + \beta_{3} + 1 ) / 3 \) (b6 + b5 + b3 + 1) / 3
\(\nu^{2}\)	\(=\)	\( ( \beta_{6} + 6\beta_{3} - 9\beta _1 + 4 ) / 3 \) (b6 + 6b3 - 9b1 + 4) / 3
\(\nu^{3}\)	\(=\)	\( ( 3\beta_{7} + 2\beta_{6} + 3\beta_{5} - 3\beta_{4} + 12\beta_{3} - 9\beta_{2} - 18\beta _1 + 5 ) / 3 \) (3b7 + 2b6 + 3b5 - 3b4 + 12b3 - 9b2 - 18*b1 + 5) / 3
\(\nu^{4}\)	\(=\)	\( ( 3\beta_{6} + 4\beta_{5} - 36\beta_{4} + 28\beta_{3} - 18\beta_{2} - 90\beta _1 - 18 ) / 3 \) (3b6 + 4b5 - 36b4 + 28b3 - 18b2 - 90b1 - 18) / 3
\(\nu^{5}\)	\(=\)	\( ( 30\beta_{7} - 40\beta_{6} + \beta_{5} - 120\beta_{4} + 46\beta_{3} - 60\beta_{2} - 210\beta _1 - 31 ) / 3 \) (30b7 - 40b6 + b5 - 120b4 + 46b3 - 60b2 - 210b1 - 31) / 3
\(\nu^{6}\)	\(=\)	\( ( 60\beta_{7} - 127\beta_{6} + 18\beta_{5} - 420\beta_{4} - 54\beta_{3} - 135\beta_{2} - 459\beta _1 - 517 ) / 3 \) (60b7 - 127b6 + 18b5 - 420b4 - 54b3 - 135b2 - 459*b1 - 517) / 3
\(\nu^{7}\)	\(=\)	\( ( 183\beta_{7} - 617\beta_{6} - 154\beta_{5} - 1128\beta_{4} - 553\beta_{3} - 126\beta_{2} - 819\beta _1 - 1538 ) / 3 \) (183b7 - 617b6 - 154b5 - 1128b4 - 553b3 - 126b2 - 819*b1 - 1538) / 3

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 107.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} + 63T^{4} + 81 \) T^8 + 63*T^4 + 81
$3$	\( T^{8} \) T^8
$5$	\( T^{8} \) T^8
$7$	\( T^{8} \) T^8
$11$	\( T^{8} \) T^8
$13$	\( T^{8} \) T^8
$17$	\( T^{8} + 2898T^{4} + 81 \) T^8 + 2898*T^4 + 81
$19$	\( (T^{4} + 98 T^{2} + 1681)^{2} \) (T^4 + 98*T^2 + 1681)^2
$23$	\( T^{8} + 9378 T^{4} + 10556001 \) T^8 + 9378*T^4 + 10556001
$29$	\( T^{8} \) T^8
$31$	\( (T^{2} - 8 T - 29)^{4} \) (T^2 - 8*T - 29)^4
$37$	\( T^{8} \) T^8
$41$	\( T^{8} \) T^8
$43$	\( T^{8} \) T^8
$47$	\( (T^{4} + 11664)^{2} \) (T^4 + 11664)^2
$53$	\( T^{8} + 13698 T^{4} + 1185921 \) T^8 + 13698*T^4 + 1185921
$59$	\( T^{8} \) T^8
$61$	\( (T^{2} + 2 T - 179)^{4} \) (T^2 + 2*T - 179)^4
$67$	\( T^{8} \) T^8
$71$	\( T^{8} \) T^8
$73$	\( T^{8} \) T^8
$79$	\( (T^{4} + 218 T^{2} + 361)^{2} \) (T^4 + 218*T^2 + 361)^2
$83$	\( T^{8} + \cdots + 3154956561 \) T^8 + 123858*T^4 + 3154956561
$89$	\( T^{8} \) T^8
$97$	\( T^{8} \) T^8
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\(n\)	\(326\)	\(352\)
\(\chi(n)\)	\(-1\)	\(-\beta_{1}\)