Newform orbit 7168.2.a.i

• Label: 7168.2.a.i
• Level: $7168$
• Weight: $2$
• Character orbit: 7168.a
• Self dual: yes
• Analytic conductor: $57.237$
• Analytic rank: $0$
• Dimension: $2$
• 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:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + 2 \beta q^{3} + 2 \beta q^{5} - q^{7} + 5 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} + 10 q^{9}+O(q^{10}) \)

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

−1.41421
1.41421

0

−2.82843

0

−2.82843

0

−1.00000

0

5.00000

0

1.2

0

2.82843

0

2.82843

0

−1.00000

0

5.00000

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.i		2
4.b	odd	2	1		7168.2.a.q		2
8.b	even	2	1	inner	7168.2.a.i		2
8.d	odd	2	1		7168.2.a.q		2
32.g	even	8	2		448.2.m.b		2
32.g	even	8	2		896.2.m.a		2
32.h	odd	8	2		112.2.m.a	✓	2
32.h	odd	8	2		896.2.m.d		2
224.x	even	8	2		784.2.m.c		2
224.be	even	24	4		784.2.x.b		4
224.bf	odd	24	4		784.2.x.g		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.2.m.a	✓	2	32.h	odd	8	2
448.2.m.b		2	32.g	even	8	2
784.2.m.c		2	224.x	even	8	2
784.2.x.b		4	224.be	even	24	4
784.2.x.g		4	224.bf	odd	24	4
896.2.m.a		2	32.g	even	8	2
896.2.m.d		2	32.h	odd	8	2
7168.2.a.i		2	1.a	even	1	1	trivial
7168.2.a.i		2	8.b	even	2	1	inner
7168.2.a.q		2	4.b	odd	2	1
7168.2.a.q		2	8.d	odd	2	1

Hecke kernels

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

\( T_{3}^{2} - 8 \)

\( T_{5}^{2} - 8 \)

\( T_{11}^{2} - 18 \)

\( T_{13} \)

\( T_{17} - 6 \)

\( T_{23} - 2 \)

\( T_{31} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 8 \)
$5$	\( T^{2} - 8 \)
$7$	\( (T + 1)^{2} \)
$11$	\( T^{2} - 18 \)