Newform orbit 9464.2.a.q

• Label: 9464.2.a.q
• Level: $9464$
• Weight: $2$
• Character orbit: 9464.a
• Self dual: yes
• Analytic conductor: $75.570$
• Analytic rank: $1$
• 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:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b * q^3 + (-b - 2) * q^5 + q^7 + (b - 2) * q^9 + (5*b - 2) * q^11 + (-3*b - 1) * q^15 - q^17 + (-3*b + 2) * q^19 + b * q^21 + 2 * q^23 + 5*b * q^25 + (-4*b + 1) * q^27 - 5*b * q^29 + (4*b - 1) * q^31 + (3*b + 5) * q^33 + (-b - 2) * q^35 + (-8*b + 4) * q^37 + (-2*b + 6) * q^41 + (-5*b + 1) * q^43 + (-b + 3) * q^45 + 3 * q^47 + q^49 - b * q^51 + (-6*b + 5) * q^53 + (-13*b - 1) * q^55 + (-b - 3) * q^57 + (4*b + 3) * q^59 - 6 * q^61 + (b - 2) * q^63 + (-4*b + 1) * q^67 + 2*b * q^69 + (-2*b + 4) * q^71 + (8*b + 2) * q^73 + (5*b + 5) * q^75 + (5*b - 2) * q^77 + (8*b - 4) * q^79 + (-6*b + 2) * q^81 + (-4*b + 3) * q^83 + (b + 2) * q^85 + (-5*b - 5) * q^87 + (-7*b - 4) * q^89 + (3*b + 4) * q^93 + (7*b - 1) * q^95 + (9*b - 11) * q^97 + (-7*b + 9) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^3 - 5 * q^5 + 2 * q^7 - 3 * q^9 + q^11 - 5 * q^15 - 2 * q^17 + q^19 + q^21 + 4 * q^23 + 5 * q^25 - 2 * q^27 - 5 * q^29 + 2 * q^31 + 13 * q^33 - 5 * q^35 + 10 * q^41 - 3 * q^43 + 5 * q^45 + 6 * q^47 + 2 * q^49 - q^51 + 4 * q^53 - 15 * q^55 - 7 * q^57 + 10 * q^59 - 12 * q^61 - 3 * q^63 - 2 * q^67 + 2 * q^69 + 6 * q^71 + 12 * q^73 + 15 * q^75 + q^77 - 2 * q^81 + 2 * q^83 + 5 * q^85 - 15 * q^87 - 15 * q^89 + 11 * q^93 + 5 * q^95 - 13 * q^97 + 11 * 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

−0.618034
1.61803

0

−0.618034

0

−1.38197

0

1.00000

0

−2.61803

0

1.2

0

1.61803

0

−3.61803

0

1.00000

0

−0.381966

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.q		2
13.b	even	2	1		9464.2.a.s		2
13.e	even	6	2		728.2.s.b	✓	4
52.i	odd	6	2		1456.2.s.n		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
728.2.s.b	✓	4	13.e	even	6	2
1456.2.s.n		4	52.i	odd	6	2
9464.2.a.q		2	1.a	even	1	1	trivial
9464.2.a.s		2	13.b	even	2	1

Hecke kernels

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

\( T_{3}^{2} - T_{3} - 1 \)

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

\( T_{11}^{2} - T_{11} - 31 \)

\( T_{17} + 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - T - 1 \)
$5$	\( T^{2} + 5T + 5 \)
$7$	\( (T - 1)^{2} \)
$11$	\( T^{2} - T - 31 \)