# Properties

 Label 9680.2.a.x Level $9680$ Weight $2$ Character orbit 9680.a Self dual yes Analytic conductor $77.295$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9680 = 2^{4} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9680.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$77.2951891566$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + 3 q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 + q^5 + 3 * q^7 - 2 * q^9 $$q + q^{3} + q^{5} + 3 q^{7} - 2 q^{9} + 6 q^{13} + q^{15} + 7 q^{17} + 5 q^{19} + 3 q^{21} + 6 q^{23} + q^{25} - 5 q^{27} - 5 q^{29} + 3 q^{31} + 3 q^{35} + 3 q^{37} + 6 q^{39} - 2 q^{41} + 4 q^{43} - 2 q^{45} + 2 q^{47} + 2 q^{49} + 7 q^{51} - q^{53} + 5 q^{57} + 10 q^{59} - 7 q^{61} - 6 q^{63} + 6 q^{65} - 8 q^{67} + 6 q^{69} - 7 q^{71} - 14 q^{73} + q^{75} + 10 q^{79} + q^{81} - 6 q^{83} + 7 q^{85} - 5 q^{87} - 15 q^{89} + 18 q^{91} + 3 q^{93} + 5 q^{95} - 12 q^{97}+O(q^{100})$$ q + q^3 + q^5 + 3 * q^7 - 2 * q^9 + 6 * q^13 + q^15 + 7 * q^17 + 5 * q^19 + 3 * q^21 + 6 * q^23 + q^25 - 5 * q^27 - 5 * q^29 + 3 * q^31 + 3 * q^35 + 3 * q^37 + 6 * q^39 - 2 * q^41 + 4 * q^43 - 2 * q^45 + 2 * q^47 + 2 * q^49 + 7 * q^51 - q^53 + 5 * q^57 + 10 * q^59 - 7 * q^61 - 6 * q^63 + 6 * q^65 - 8 * q^67 + 6 * q^69 - 7 * q^71 - 14 * q^73 + q^75 + 10 * q^79 + q^81 - 6 * q^83 + 7 * q^85 - 5 * q^87 - 15 * q^89 + 18 * q^91 + 3 * q^93 + 5 * q^95 - 12 * q^97

## 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 1.00000 0 3.00000 0 −2.00000 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$-1$$

## 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 9680.2.a.x 1
4.b odd 2 1 1210.2.a.b 1
11.b odd 2 1 880.2.a.i 1
20.d odd 2 1 6050.2.a.bj 1
33.d even 2 1 7920.2.a.d 1
44.c even 2 1 110.2.a.b 1
55.d odd 2 1 4400.2.a.l 1
55.e even 4 2 4400.2.b.i 2
88.b odd 2 1 3520.2.a.h 1
88.g even 2 1 3520.2.a.y 1
132.d odd 2 1 990.2.a.d 1
220.g even 2 1 550.2.a.f 1
220.i odd 4 2 550.2.b.a 2
308.g odd 2 1 5390.2.a.bf 1
660.g odd 2 1 4950.2.a.bc 1
660.q even 4 2 4950.2.c.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.b 1 44.c even 2 1
550.2.a.f 1 220.g even 2 1
550.2.b.a 2 220.i odd 4 2
880.2.a.i 1 11.b odd 2 1
990.2.a.d 1 132.d odd 2 1
1210.2.a.b 1 4.b odd 2 1
3520.2.a.h 1 88.b odd 2 1
3520.2.a.y 1 88.g even 2 1
4400.2.a.l 1 55.d odd 2 1
4400.2.b.i 2 55.e even 4 2
4950.2.a.bc 1 660.g odd 2 1
4950.2.c.m 2 660.q even 4 2
5390.2.a.bf 1 308.g odd 2 1
6050.2.a.bj 1 20.d odd 2 1
7920.2.a.d 1 33.d even 2 1
9680.2.a.x 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9680))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} - 3$$ T7 - 3 $$T_{13} - 6$$ T13 - 6 $$T_{17} - 7$$ T17 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T - 3$$
$11$ $$T$$
$13$ $$T - 6$$
$17$ $$T - 7$$
$19$ $$T - 5$$
$23$ $$T - 6$$
$29$ $$T + 5$$
$31$ $$T - 3$$
$37$ $$T - 3$$
$41$ $$T + 2$$
$43$ $$T - 4$$
$47$ $$T - 2$$
$53$ $$T + 1$$
$59$ $$T - 10$$
$61$ $$T + 7$$
$67$ $$T + 8$$
$71$ $$T + 7$$
$73$ $$T + 14$$
$79$ $$T - 10$$
$83$ $$T + 6$$
$89$ $$T + 15$$
$97$ $$T + 12$$