Newform orbit 9408.2.a.dh

• Label: 9408.2.a.dh
• Level: $9408$
• Weight: $2$
• Character orbit: 9408.a
• Self dual: yes
• Analytic conductor: $75.123$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9408.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(75.1232582216\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 1176)
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.

\(f(q)\)	\(=\)	\( q - q^{3} + (\beta - 2) q^{5} + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) \)

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

−1.00000

0

−3.41421

0

0

0

1.00000

0

1.2

0

−1.00000

0

−0.585786

0

0

0

1.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(3\)	\(1\)
\(7\)	\(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	9408.2.a.dh		2
4.b	odd	2	1		9408.2.a.dv		2
7.b	odd	2	1		9408.2.a.ed		2
8.b	even	2	1		2352.2.a.bg		2
8.d	odd	2	1		1176.2.a.l	✓	2
24.f	even	2	1		3528.2.a.bc		2
24.h	odd	2	1		7056.2.a.ce		2
28.d	even	2	1		9408.2.a.dr		2
56.e	even	2	1		1176.2.a.m	yes	2
56.h	odd	2	1		2352.2.a.z		2
56.j	odd	6	2		2352.2.q.bg		4
56.k	odd	6	2		1176.2.q.n		4
56.m	even	6	2		1176.2.q.m		4
56.p	even	6	2		2352.2.q.ba		4
168.e	odd	2	1		3528.2.a.bm		2
168.i	even	2	1		7056.2.a.cw		2
168.v	even	6	2		3528.2.s.bl		4
168.be	odd	6	2		3528.2.s.bc		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1176.2.a.l	✓	2	8.d	odd	2	1
1176.2.a.m	yes	2	56.e	even	2	1
1176.2.q.m		4	56.m	even	6	2
1176.2.q.n		4	56.k	odd	6	2
2352.2.a.z		2	56.h	odd	2	1
2352.2.a.bg		2	8.b	even	2	1
2352.2.q.ba		4	56.p	even	6	2
2352.2.q.bg		4	56.j	odd	6	2
3528.2.a.bc		2	24.f	even	2	1
3528.2.a.bm		2	168.e	odd	2	1
3528.2.s.bc		4	168.be	odd	6	2
3528.2.s.bl		4	168.v	even	6	2
7056.2.a.ce		2	24.h	odd	2	1
7056.2.a.cw		2	168.i	even	2	1
9408.2.a.dh		2	1.a	even	1	1	trivial
9408.2.a.dr		2	28.d	even	2	1
9408.2.a.dv		2	4.b	odd	2	1
9408.2.a.ed		2	7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5}^{2} + 4T_{5} + 2 \)

\( T_{11}^{2} + 4T_{11} - 4 \)

\( T_{13}^{2} - 18 \)

\( T_{17}^{2} - 12T_{17} + 34 \)

\( T_{19}^{2} - 8T_{19} + 8 \)

\( T_{31}^{2} - 8 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T + 1)^{2} \)
$5$	\( T^{2} + 4T + 2 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 4T - 4 \)