# Properties

 Label 9600.2.a.bh Level $9600$ Weight $2$ Character orbit 9600.a Self dual yes Analytic conductor $76.656$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9600 = 2^{7} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9600.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$76.6563859404$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - 2q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} - 2q^{7} + q^{9} - 4q^{11} - 6q^{13} - 6q^{17} - 2q^{21} - 4q^{23} + q^{27} + 4q^{29} + 10q^{31} - 4q^{33} - 2q^{37} - 6q^{39} - 2q^{41} - 8q^{43} + 12q^{47} - 3q^{49} - 6q^{51} + 12q^{53} - 4q^{59} + 2q^{61} - 2q^{63} - 4q^{67} - 4q^{69} - 4q^{71} + 10q^{73} + 8q^{77} - 6q^{79} + q^{81} - 12q^{83} + 4q^{87} + 2q^{89} + 12q^{91} + 10q^{93} + 6q^{97} - 4q^{99} + O(q^{100})$$

## 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 1.00000 0 0 0 −2.00000 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$

## 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 9600.2.a.bh 1
4.b odd 2 1 9600.2.a.w 1
5.b even 2 1 384.2.a.c yes 1
8.b even 2 1 9600.2.a.h 1
8.d odd 2 1 9600.2.a.bw 1
15.d odd 2 1 1152.2.a.l 1
20.d odd 2 1 384.2.a.f yes 1
40.e odd 2 1 384.2.a.b 1
40.f even 2 1 384.2.a.g yes 1
60.h even 2 1 1152.2.a.i 1
80.k odd 4 2 768.2.d.g 2
80.q even 4 2 768.2.d.b 2
120.i odd 2 1 1152.2.a.k 1
120.m even 2 1 1152.2.a.j 1
240.t even 4 2 2304.2.d.m 2
240.bm odd 4 2 2304.2.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.b 1 40.e odd 2 1
384.2.a.c yes 1 5.b even 2 1
384.2.a.f yes 1 20.d odd 2 1
384.2.a.g yes 1 40.f even 2 1
768.2.d.b 2 80.q even 4 2
768.2.d.g 2 80.k odd 4 2
1152.2.a.i 1 60.h even 2 1
1152.2.a.j 1 120.m even 2 1
1152.2.a.k 1 120.i odd 2 1
1152.2.a.l 1 15.d odd 2 1
2304.2.d.d 2 240.bm odd 4 2
2304.2.d.m 2 240.t even 4 2
9600.2.a.h 1 8.b even 2 1
9600.2.a.w 1 4.b odd 2 1
9600.2.a.bh 1 1.a even 1 1 trivial
9600.2.a.bw 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9600))$$:

 $$T_{7} + 2$$ $$T_{11} + 4$$ $$T_{13} + 6$$ $$T_{17} + 6$$ $$T_{29} - 4$$ $$T_{31} - 10$$

## Hecke characteristic polynomials

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