Newform orbit 9248.2.a.k

• Label: 9248.2.a.k
• Level: $9248$
• Weight: $2$
• Character orbit: 9248.a
• Self dual: yes
• Analytic conductor: $73.846$
• Analytic rank: $1$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9248 = 2^{5} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9248.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(73.8456517893\)
Analytic rank:	\(1\)
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 544)
Fricke sign:	\(+1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-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 - b * q^5 - 3 * q^9 + 4 * q^13 - 3 * q^25 - 3*b * q^29 + 5*b * q^37 + 9*b * q^41 + 3*b * q^45 - 7 * q^49 - 4 * q^53 - b * q^61 - 4*b * q^65 + 5*b * q^73 + 9 * q^81 - 16 * q^89 - 5*b * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9} + 8 q^{13} - 6 q^{25} - 14 q^{49} - 8 q^{53} + 18 q^{81} - 32 q^{89}+O(q^{100}) \)

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

0

0

−1.41421

0

0

0

−3.00000

0

1.2

0

0

0

1.41421

0

0

0

−3.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(17\)	\( +1 \)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
17.b	even	2	1	inner
68.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	9248.2.a.k		2
4.b	odd	2	1	CM	9248.2.a.k		2
17.b	even	2	1	inner	9248.2.a.k		2
17.d	even	8	2		544.2.o.c	✓	2
68.d	odd	2	1	inner	9248.2.a.k		2
68.g	odd	8	2		544.2.o.c	✓	2
136.o	even	8	2		1088.2.o.j		2
136.p	odd	8	2		1088.2.o.j		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
544.2.o.c	✓	2	17.d	even	8	2
544.2.o.c	✓	2	68.g	odd	8	2
1088.2.o.j		2	136.o	even	8	2
1088.2.o.j		2	136.p	odd	8	2
9248.2.a.k		2	1.a	even	1	1	trivial
9248.2.a.k		2	4.b	odd	2	1	CM
9248.2.a.k		2	17.b	even	2	1	inner
9248.2.a.k		2	68.d	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9248))\):

\( T_{3} \)

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

\( T_{7} \)

\( T_{19} \)

\( T_{43} \)

Hecke characteristic polynomials

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