Newform orbit 9464.2.a.k

• Label: 9464.2.a.k
• Level: $9464$
• Weight: $2$
• Character orbit: 9464.a
• Self dual: yes
• Analytic conductor: $75.570$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9464.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(75.5704204729\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 728)
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 + (b - 2) * q^3 + (-b + 1) * q^5 - q^7 + (-4*b + 3) * q^9 + (3*b + 2) * q^11 + (3*b - 4) * q^15 + (-b + 2) * q^17 + (b + 5) * q^19 + (-b + 2) * q^21 + (4*b - 1) * q^23 + (-2*b - 2) * q^25 + (8*b - 8) * q^27 + (-2*b + 3) * q^29 + (-b + 7) * q^31 + (-4*b + 2) * q^33 + (b - 1) * q^35 + (-3*b + 4) * q^37 + (2*b - 6) * q^41 + (-6*b - 3) * q^43 + (-7*b + 11) * q^45 + (3*b - 1) * q^47 + q^49 + (4*b - 6) * q^51 + (6*b + 1) * q^53 + (b - 4) * q^55 + (3*b - 8) * q^57 + (4*b + 6) * q^59 - 10 * q^61 + (4*b - 3) * q^63 + (-2*b + 6) * q^67 + (-9*b + 10) * q^69 + (-b + 4) * q^71 + (-3*b + 5) * q^73 + 2*b * q^75 + (-3*b - 2) * q^77 + (-4*b + 1) * q^79 + (-12*b + 23) * q^81 + (-5*b - 7) * q^83 + (-3*b + 4) * q^85 + (7*b - 10) * q^87 + (-b + 1) * q^89 + (9*b - 16) * q^93 + (-4*b + 3) * q^95 + (-7*b + 9) * q^97 + (b - 18) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^3 + 2 * q^5 - 2 * q^7 + 6 * q^9 + 4 * q^11 - 8 * q^15 + 4 * q^17 + 10 * q^19 + 4 * q^21 - 2 * q^23 - 4 * q^25 - 16 * q^27 + 6 * q^29 + 14 * q^31 + 4 * q^33 - 2 * q^35 + 8 * q^37 - 12 * q^41 - 6 * q^43 + 22 * q^45 - 2 * q^47 + 2 * q^49 - 12 * q^51 + 2 * q^53 - 8 * q^55 - 16 * q^57 + 12 * q^59 - 20 * q^61 - 6 * q^63 + 12 * q^67 + 20 * q^69 + 8 * q^71 + 10 * q^73 - 4 * q^77 + 2 * q^79 + 46 * q^81 - 14 * q^83 + 8 * q^85 - 20 * q^87 + 2 * q^89 - 32 * q^93 + 6 * q^95 + 18 * q^97 - 36 * q^99

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

−3.41421

0

2.41421

0

−1.00000

0

8.65685

0

1.2

0

−0.585786

0

−0.414214

0

−1.00000

0

−2.65685

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(7\)	\( +1 \)
\(13\)	\( +1 \)

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	9464.2.a.k		2
13.b	even	2	1		728.2.a.e	✓	2
39.d	odd	2	1		6552.2.a.bj		2
52.b	odd	2	1		1456.2.a.s		2
91.b	odd	2	1		5096.2.a.q		2
104.e	even	2	1		5824.2.a.br		2
104.h	odd	2	1		5824.2.a.bg		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
728.2.a.e	✓	2	13.b	even	2	1
1456.2.a.s		2	52.b	odd	2	1
5096.2.a.q		2	91.b	odd	2	1
5824.2.a.bg		2	104.h	odd	2	1
5824.2.a.br		2	104.e	even	2	1
6552.2.a.bj		2	39.d	odd	2	1
9464.2.a.k		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9464))\):

\( T_{3}^{2} + 4T_{3} + 2 \)

\( T_{5}^{2} - 2T_{5} - 1 \)

\( T_{11}^{2} - 4T_{11} - 14 \)

\( T_{17}^{2} - 4T_{17} + 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 4T + 2 \)
$5$	\( T^{2} - 2T - 1 \)
$7$	\( (T + 1)^{2} \)
$11$	\( T^{2} - 4T - 14 \)