Newform orbit 7168.2.a.bj

• Label: 7168.2.a.bj
• Level: $7168$
• Weight: $2$
• Character orbit: 7168.a
• Self dual: yes
• Analytic conductor: $57.237$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(57.2367681689\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - 24x^{10} + 221x^{8} - 968x^{6} + 2008x^{4} - 1640x^{2} + 196 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{7} \)
Twist minimal:	no (minimal twist has level 112)
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.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + (\beta_{4} + \beta_{2} + \beta_1) q^{5} + q^{7} + (\beta_{9} + \beta_{8} + 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7} + 12 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 24x^{10} + 221x^{8} - 968x^{6} + 2008x^{4} - 1640x^{2} + 196 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{11} + 17\nu^{9} - 95\nu^{7} + 177\nu^{5} - 6\nu^{3} - 68\nu ) / 42 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{11} + 17\nu^{9} - 95\nu^{7} + 177\nu^{5} + 36\nu^{3} - 278\nu ) / 42 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{8} - 16\nu^{6} + 79\nu^{4} - 116\nu^{2} + 16 ) / 6 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{11} + 24\nu^{9} - 207\nu^{7} + 772\nu^{5} - 1154\nu^{3} + 464\nu ) / 84 \)
\(\beta_{5}\)	\(=\)	\( ( 2\nu^{11} - 41\nu^{9} + 316\nu^{7} - 1117\nu^{5} + 1720\nu^{3} - 830\nu ) / 42 \)
\(\beta_{6}\)	\(=\)	\( ( -\nu^{11} + 18\nu^{9} - 115\nu^{7} + 322\nu^{5} - 426\nu^{3} + 264\nu ) / 12 \)
\(\beta_{7}\)	\(=\)	\( ( -\nu^{10} + 18\nu^{8} - 113\nu^{6} + 286\nu^{4} - 232\nu^{2} - 26 ) / 6 \)
\(\beta_{8}\)	\(=\)	\( ( -\nu^{10} + 18\nu^{8} - 113\nu^{6} + 286\nu^{4} - 244\nu^{2} + 22 ) / 6 \)
\(\beta_{9}\)	\(=\)	\( ( \nu^{10} - 18\nu^{8} + 115\nu^{6} - 310\nu^{4} + 318\nu^{2} - 54 ) / 6 \)
\(\beta_{10}\)	\(=\)	\( ( -\nu^{10} + 19\nu^{8} - 127\nu^{6} + 353\nu^{4} - 370\nu^{2} + 72 ) / 6 \)
\(\beta_{11}\)	\(=\)	\( ( 15\nu^{11} - 276\nu^{9} + 1789\nu^{7} - 4776\nu^{5} + 4402\nu^{3} + 152\nu ) / 84 \)

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

−1.33191
−1.75510
2.66376
−2.48658
2.39251
0.377920
−0.377920
−2.39251
2.48658
−2.66376
1.75510
1.33191

0

−3.13347

0

−0.555963

0

1.00000

0

6.81864

0

1.2

0

−2.90620

0

−3.85750

0

1.00000

0

5.44602

0

1.3

0

−1.96750

0

3.06146

0

1.00000

0

0.871066

0

1.4

0

−0.892634

0

−3.31293

0

1.00000

0

−2.20320

0

1.5

0

−0.848497

0

−1.37882

0

1.00000

0

−2.28005

0

1.6

0

−0.589521

0

−1.60045

0

1.00000

0

−2.65247

0

1.7

0

0.589521

0

1.60045

0

1.00000

0

−2.65247

0

1.8

0

0.848497

0

1.37882

0

1.00000

0

−2.28005

0

1.9

0

0.892634

0

3.31293

0

1.00000

0

−2.20320

0

1.10

0

1.96750

0

−3.06146

0

1.00000

0

0.871066

0

1.11

0

2.90620

0

3.85750

0

1.00000

0

5.44602

0

1.12

0

3.13347

0

0.555963

0

1.00000

0

6.81864

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(7\)	\(-1\)

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7168.2.a.bj		12
4.b	odd	2	1		7168.2.a.bi		12
8.b	even	2	1	inner	7168.2.a.bj		12
8.d	odd	2	1		7168.2.a.bi		12
32.g	even	8	2		112.2.m.d	✓	12
32.g	even	8	2		896.2.m.g		12
32.h	odd	8	2		448.2.m.d		12
32.h	odd	8	2		896.2.m.h		12
224.v	odd	8	2		784.2.m.h		12
224.bc	odd	24	4		784.2.x.m		24
224.bd	even	24	4		784.2.x.l		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.2.m.d	✓	12	32.g	even	8	2
448.2.m.d		12	32.h	odd	8	2
784.2.m.h		12	224.v	odd	8	2
784.2.x.l		24	224.bd	even	24	4
784.2.x.m		24	224.bc	odd	24	4
896.2.m.g		12	32.g	even	8	2
896.2.m.h		12	32.h	odd	8	2
7168.2.a.bi		12	4.b	odd	2	1
7168.2.a.bi		12	8.d	odd	2	1
7168.2.a.bj		12	1.a	even	1	1	trivial
7168.2.a.bj		12	8.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7168))\):

\( T_{3}^{12} - 24T_{3}^{10} + 196T_{3}^{8} - 632T_{3}^{6} + 772T_{3}^{4} - 384T_{3}^{2} + 64 \)

\( T_{5}^{12} - 40T_{5}^{10} + 580T_{5}^{8} - 3688T_{5}^{6} + 9892T_{5}^{4} - 10176T_{5}^{2} + 2304 \)

\( T_{11}^{12} - 76T_{11}^{10} + 2148T_{11}^{8} - 28736T_{11}^{6} + 190080T_{11}^{4} - 569344T_{11}^{2} + 541696 \)

\( T_{13}^{12} - 112T_{13}^{10} + 4548T_{13}^{8} - 82856T_{13}^{6} + 694020T_{13}^{4} - 2563072T_{13}^{2} + 3211264 \)

\( T_{17}^{6} - 4T_{17}^{5} - 44T_{17}^{4} + 200T_{17}^{3} + 40T_{17}^{2} - 288T_{17} - 96 \)

\( T_{23}^{6} - 8T_{23}^{5} - 36T_{23}^{4} + 320T_{23}^{3} + 340T_{23}^{2} - 2656T_{23} - 3184 \)

\( T_{31}^{6} + 4T_{31}^{5} - 16T_{31}^{4} - 72T_{31}^{3} - 24T_{31}^{2} + 96T_{31} + 64 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{12} \)
$3$	T^12 - 24*T^10 + 196*T^8 - 632*T^6 + 772*T^4 - 384*T^2 + 64
$5$	T^12 - 40*T^10 + 580*T^8 - 3688*T^6 + 9892*T^4 - 10176*T^2 + 2304
$7$	\( (T - 1)^{12} \)
$11$	T^12 - 76*T^10 + 2148*T^8 - 28736*T^6 + 190080*T^4 - 569344*T^2 + 541696