# Properties

 Label 9360.2.a.bh Level $9360$ Weight $2$ Character orbit 9360.a Self dual yes Analytic conductor $74.740$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9360.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$74.7399762919$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} - 2q^{7} + O(q^{10})$$ $$q + q^{5} - 2q^{7} + 4q^{11} - q^{13} - 8q^{17} + 6q^{19} + 6q^{23} + q^{25} + 4q^{29} - 2q^{35} - 2q^{37} + 2q^{41} + 4q^{43} - 3q^{49} + 10q^{53} + 4q^{55} + 4q^{59} - 10q^{61} - q^{65} - 12q^{67} - 8q^{71} - 8q^{73} - 8q^{77} - 8q^{79} + 12q^{83} - 8q^{85} + 14q^{89} + 2q^{91} + 6q^{95} - 16q^{97} + O(q^{100})$$

## 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
 0
0 0 0 1.00000 0 −2.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$1$$

## 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 9360.2.a.bh 1
3.b odd 2 1 3120.2.a.o 1
4.b odd 2 1 1170.2.a.e 1
12.b even 2 1 390.2.a.e 1
20.d odd 2 1 5850.2.a.bi 1
20.e even 4 2 5850.2.e.i 2
60.h even 2 1 1950.2.a.h 1
60.l odd 4 2 1950.2.e.f 2
156.h even 2 1 5070.2.a.e 1
156.l odd 4 2 5070.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.e 1 12.b even 2 1
1170.2.a.e 1 4.b odd 2 1
1950.2.a.h 1 60.h even 2 1
1950.2.e.f 2 60.l odd 4 2
3120.2.a.o 1 3.b odd 2 1
5070.2.a.e 1 156.h even 2 1
5070.2.b.e 2 156.l odd 4 2
5850.2.a.bi 1 20.d odd 2 1
5850.2.e.i 2 20.e even 4 2
9360.2.a.bh 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9360))$$:

 $$T_{7} + 2$$ $$T_{11} - 4$$ $$T_{17} + 8$$ $$T_{19} - 6$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$2 + T$$
$11$ $$-4 + T$$
$13$ $$1 + T$$
$17$ $$8 + T$$
$19$ $$-6 + T$$
$23$ $$-6 + T$$
$29$ $$-4 + T$$
$31$ $$T$$
$37$ $$2 + T$$
$41$ $$-2 + T$$
$43$ $$-4 + T$$
$47$ $$T$$
$53$ $$-10 + T$$
$59$ $$-4 + T$$
$61$ $$10 + T$$
$67$ $$12 + T$$
$71$ $$8 + T$$
$73$ $$8 + T$$
$79$ $$8 + T$$
$83$ $$-12 + T$$
$89$ $$-14 + T$$
$97$ $$16 + T$$