Newform orbit 8281.2.a.co

• Label: 8281.2.a.co
• Level: $8281$
• Weight: $2$
• Character orbit: 8281.a
• Self dual: yes
• Analytic conductor: $66.124$
• Analytic rank: $1$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(66.1241179138\)
Analytic rank:	\(1\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - 16x^{10} + 88x^{8} - 197x^{6} + 172x^{4} - 36x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b1 * q^2 + b3 * q^3 + (b9 - b8 + 1) * q^4 + (-b11 - b1) * q^5 + b4 * q^6 + (-b11 + b10 + b7 + b5 + b4 + b1) * q^8 + (-b3 + b2) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{3} + 8 q^{4} + 2 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{10} + 88x^{8} - 197x^{6} + 172x^{4} - 36x^{2} + 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( 5\nu^{10} - 66\nu^{8} + 242\nu^{6} - 127\nu^{4} - 248\nu^{2} + 32 ) / 11 \)
\(\beta_{3}\)	\(=\)	\( ( -3\nu^{10} + 44\nu^{8} - 209\nu^{6} + 360\nu^{4} - 223\nu^{2} + 27 ) / 11 \)
\(\beta_{4}\)	\(=\)	\( ( -3\nu^{11} + 44\nu^{9} - 209\nu^{7} + 360\nu^{5} - 223\nu^{3} + 27\nu ) / 11 \)
\(\beta_{5}\)	\(=\)	\( ( -2\nu^{11} + 22\nu^{9} - 33\nu^{7} - 244\nu^{5} + 559\nu^{3} - 191\nu ) / 11 \)
\(\beta_{6}\)	\(=\)	\( ( -8\nu^{10} + 110\nu^{8} - 451\nu^{6} + 476\nu^{4} + 91\nu^{2} - 27 ) / 11 \)
\(\beta_{7}\)	\(=\)	\( \nu^{9} - 14\nu^{7} + 60\nu^{5} - 76\nu^{3} + 12\nu \)
\(\beta_{8}\)	\(=\)	\( ( 10\nu^{10} - 143\nu^{8} + 638\nu^{6} - 903\nu^{4} + 263\nu^{2} + 9 ) / 11 \)
\(\beta_{9}\)	\(=\)	\( ( 10\nu^{10} - 143\nu^{8} + 638\nu^{6} - 903\nu^{4} + 274\nu^{2} - 24 ) / 11 \)
\(\beta_{10}\)	\(=\)	\( ( -8\nu^{11} + 121\nu^{9} - 605\nu^{7} + 1147\nu^{5} - 822\nu^{3} + 171\nu ) / 11 \)
\(\beta_{11}\)	\(=\)	\( ( -13\nu^{11} + 198\nu^{9} - 1001\nu^{7} + 1923\nu^{5} - 1333\nu^{3} + 194\nu ) / 11 \)

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.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−2.58860
−2.30327
−1.37905
−1.34523
−0.499987
−0.180824
0.180824
0.499987
1.34523
1.37905
2.30327
2.58860

−2.58860

−0.518466

4.70085

1.61205

1.34210

0

−6.99143

−2.73119

−4.17296

1.2

−2.30327

1.47336

3.30504

0.847292

−3.39354

0

−3.00585

−0.829208

−1.95154

1.3

−1.37905

−2.88120

−0.0982074

0.805948

3.97334

0

2.89354

5.30133

−1.11145

1.4

−1.34523

−2.05010

−0.190366

3.56778

2.75785

0

2.94654

1.20292

−4.79947

1.5

−0.499987

−0.849601

−1.75001

−1.04248

0.424789

0

1.87496

−2.27818

0.521224

1.6

−0.180824

1.82601

−1.96730

2.68664

−0.330186

0

0.717383

0.334323

−0.485809

1.7

0.180824

1.82601

−1.96730

−2.68664

0.330186

0

−0.717383

0.334323

−0.485809

1.8

0.499987

−0.849601

−1.75001

1.04248

−0.424789

0

−1.87496

−2.27818

0.521224

1.9

1.34523

−2.05010

−0.190366

−3.56778

−2.75785

0

−2.94654

1.20292

−4.79947

1.10

1.37905

−2.88120

−0.0982074

−0.805948

−3.97334

0

−2.89354

5.30133

−1.11145

1.11

2.30327

1.47336

3.30504

−0.847292

3.39354

0

3.00585

−0.829208

−1.95154

1.12

2.58860

−0.518466

4.70085

−1.61205

−1.34210

0

6.99143

−2.73119

−4.17296

Atkin-Lehner signs

\( p \)	Sign
\(7\)	\(-1\)
\(13\)	\(-1\)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8281.2.a.co		12
7.b	odd	2	1		8281.2.a.cp		12
7.d	odd	6	2		1183.2.e.j		24
13.b	even	2	1	inner	8281.2.a.co		12
13.f	odd	12	2		637.2.q.i		12
91.b	odd	2	1		8281.2.a.cp		12
91.s	odd	6	2		1183.2.e.j		24
91.w	even	12	2		91.2.u.b	yes	12
91.x	odd	12	2		637.2.k.i		12
91.ba	even	12	2		91.2.k.b	✓	12
91.bc	even	12	2		637.2.q.g		12
91.bd	odd	12	2		637.2.u.g		12
273.bs	odd	12	2		819.2.bm.f		12
273.ch	odd	12	2		819.2.do.e		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
91.2.k.b	✓	12	91.ba	even	12	2
91.2.u.b	yes	12	91.w	even	12	2
637.2.k.i		12	91.x	odd	12	2
637.2.q.g		12	91.bc	even	12	2
637.2.q.i		12	13.f	odd	12	2
637.2.u.g		12	91.bd	odd	12	2
819.2.bm.f		12	273.bs	odd	12	2
819.2.do.e		12	273.ch	odd	12	2
1183.2.e.j		24	7.d	odd	6	2
1183.2.e.j		24	91.s	odd	6	2
8281.2.a.co		12	1.a	even	1	1	trivial
8281.2.a.co		12	13.b	even	2	1	inner
8281.2.a.cp		12	7.b	odd	2	1
8281.2.a.cp		12	91.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{12} - 16T_{2}^{10} + 88T_{2}^{8} - 197T_{2}^{6} + 172T_{2}^{4} - 36T_{2}^{2} + 1 \)

\( T_{3}^{6} + 3T_{3}^{5} - 5T_{3}^{4} - 16T_{3}^{3} + 4T_{3}^{2} + 19T_{3} + 7 \)

\( T_{5}^{12} - 25T_{5}^{10} + 201T_{5}^{8} - 636T_{5}^{6} + 878T_{5}^{4} - 539T_{5}^{2} + 121 \)

\( T_{11}^{12} - 62T_{11}^{10} + 1355T_{11}^{8} - 13284T_{11}^{6} + 61227T_{11}^{4} - 122593T_{11}^{2} + 85849 \)

\( T_{17}^{6} + 17T_{17}^{5} + 96T_{17}^{4} + 198T_{17}^{3} + 56T_{17}^{2} - 146T_{17} + 19 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^12 - 16*T^10 + 88*T^8 - 197*T^6 + 172*T^4 - 36*T^2 + 1
$3$	(T^6 + 3*T^5 - 5*T^4 - 16*T^3 + 4*T^2 + 19*T + 7)^2
$5$	T^12 - 25*T^10 + 201*T^8 - 636*T^6 + 878*T^4 - 539*T^2 + 121
$7$	\( T^{12} \)
$11$	T^12 - 62*T^10 + 1355*T^8 - 13284*T^6 + 61227*T^4 - 122593*T^2 + 85849