# Properties

 Label 9680.2.a.r 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 55) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} - 3 q^{9}+O(q^{10})$$ q + q^5 - 3 * q^9 $$q + q^{5} - 3 q^{9} - 2 q^{13} - 6 q^{17} - 4 q^{19} - 4 q^{23} + q^{25} - 6 q^{29} + 8 q^{31} - 2 q^{37} - 2 q^{41} + 4 q^{43} - 3 q^{45} + 12 q^{47} - 7 q^{49} - 2 q^{53} - 4 q^{59} + 10 q^{61} - 2 q^{65} + 16 q^{67} - 8 q^{71} - 14 q^{73} + 8 q^{79} + 9 q^{81} - 4 q^{83} - 6 q^{85} + 10 q^{89} - 4 q^{95} + 10 q^{97}+O(q^{100})$$ q + q^5 - 3 * q^9 - 2 * q^13 - 6 * q^17 - 4 * q^19 - 4 * q^23 + q^25 - 6 * q^29 + 8 * q^31 - 2 * q^37 - 2 * q^41 + 4 * q^43 - 3 * q^45 + 12 * q^47 - 7 * q^49 - 2 * q^53 - 4 * q^59 + 10 * q^61 - 2 * q^65 + 16 * q^67 - 8 * q^71 - 14 * q^73 + 8 * q^79 + 9 * q^81 - 4 * q^83 - 6 * q^85 + 10 * q^89 - 4 * q^95 + 10 * 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 0 0 1.00000 0 0 0 −3.00000 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.r 1
4.b odd 2 1 605.2.a.b 1
11.b odd 2 1 880.2.a.h 1
12.b even 2 1 5445.2.a.i 1
20.d odd 2 1 3025.2.a.f 1
33.d even 2 1 7920.2.a.i 1
44.c even 2 1 55.2.a.a 1
44.g even 10 4 605.2.g.a 4
44.h odd 10 4 605.2.g.c 4
55.d odd 2 1 4400.2.a.p 1
55.e even 4 2 4400.2.b.n 2
88.b odd 2 1 3520.2.a.n 1
88.g even 2 1 3520.2.a.p 1
132.d odd 2 1 495.2.a.a 1
220.g even 2 1 275.2.a.a 1
220.i odd 4 2 275.2.b.b 2
308.g odd 2 1 2695.2.a.c 1
572.b even 2 1 9295.2.a.b 1
660.g odd 2 1 2475.2.a.i 1
660.q even 4 2 2475.2.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.a 1 44.c even 2 1
275.2.a.a 1 220.g even 2 1
275.2.b.b 2 220.i odd 4 2
495.2.a.a 1 132.d odd 2 1
605.2.a.b 1 4.b odd 2 1
605.2.g.a 4 44.g even 10 4
605.2.g.c 4 44.h odd 10 4
880.2.a.h 1 11.b odd 2 1
2475.2.a.i 1 660.g odd 2 1
2475.2.c.f 2 660.q even 4 2
2695.2.a.c 1 308.g odd 2 1
3025.2.a.f 1 20.d odd 2 1
3520.2.a.n 1 88.b odd 2 1
3520.2.a.p 1 88.g even 2 1
4400.2.a.p 1 55.d odd 2 1
4400.2.b.n 2 55.e even 4 2
5445.2.a.i 1 12.b even 2 1
7920.2.a.i 1 33.d even 2 1
9295.2.a.b 1 572.b even 2 1
9680.2.a.r 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9680))$$:

 $$T_{3}$$ T3 $$T_{7}$$ T7 $$T_{13} + 2$$ T13 + 2 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

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