# Properties

 Label 9680.2.a.a Level $9680$ Weight $2$ Character orbit 9680.a Self dual yes Analytic conductor $77.295$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + q^{5} + q^{7} + 6 q^{9} + O(q^{10})$$ $$q - 3 q^{3} + q^{5} + q^{7} + 6 q^{9} + 6 q^{13} - 3 q^{15} - 3 q^{17} - 5 q^{19} - 3 q^{21} + 2 q^{23} + q^{25} - 9 q^{27} + 5 q^{29} - 5 q^{31} + q^{35} - q^{37} - 18 q^{39} + 2 q^{41} + 12 q^{43} + 6 q^{45} + 2 q^{47} - 6 q^{49} + 9 q^{51} - 13 q^{53} + 15 q^{57} - 2 q^{59} - q^{61} + 6 q^{63} + 6 q^{65} - 16 q^{67} - 6 q^{69} - 15 q^{71} - 10 q^{73} - 3 q^{75} + 2 q^{79} + 9 q^{81} - 14 q^{83} - 3 q^{85} - 15 q^{87} + 9 q^{89} + 6 q^{91} + 15 q^{93} - 5 q^{95} - 16 q^{97} + O(q^{100})$$

## 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 −3.00000 0 1.00000 0 1.00000 0 6.00000 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.a 1
4.b odd 2 1 4840.2.a.i 1
11.b odd 2 1 880.2.a.a 1
33.d even 2 1 7920.2.a.e 1
44.c even 2 1 440.2.a.d 1
55.d odd 2 1 4400.2.a.be 1
55.e even 4 2 4400.2.b.a 2
88.b odd 2 1 3520.2.a.bh 1
88.g even 2 1 3520.2.a.a 1
132.d odd 2 1 3960.2.a.f 1
220.g even 2 1 2200.2.a.a 1
220.i odd 4 2 2200.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.d 1 44.c even 2 1
880.2.a.a 1 11.b odd 2 1
2200.2.a.a 1 220.g even 2 1
2200.2.b.b 2 220.i odd 4 2
3520.2.a.a 1 88.g even 2 1
3520.2.a.bh 1 88.b odd 2 1
3960.2.a.f 1 132.d odd 2 1
4400.2.a.be 1 55.d odd 2 1
4400.2.b.a 2 55.e even 4 2
4840.2.a.i 1 4.b odd 2 1
7920.2.a.e 1 33.d even 2 1
9680.2.a.a 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9680))$$:

 $$T_{3} + 3$$ $$T_{7} - 1$$ $$T_{13} - 6$$ $$T_{17} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$-1 + T$$
$7$ $$-1 + T$$
$11$ $$T$$
$13$ $$-6 + T$$
$17$ $$3 + T$$
$19$ $$5 + T$$
$23$ $$-2 + T$$
$29$ $$-5 + T$$
$31$ $$5 + T$$
$37$ $$1 + T$$
$41$ $$-2 + T$$
$43$ $$-12 + T$$
$47$ $$-2 + T$$
$53$ $$13 + T$$
$59$ $$2 + T$$
$61$ $$1 + T$$
$67$ $$16 + T$$
$71$ $$15 + T$$
$73$ $$10 + T$$
$79$ $$-2 + T$$
$83$ $$14 + T$$
$89$ $$-9 + T$$
$97$ $$16 + T$$