# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 605) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + q^{5} + \beta q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^3 + q^5 + b * q^7 + q^9 $$q - 2 q^{3} + q^{5} + \beta q^{7} + q^{9} - 2 q^{15} + 2 \beta q^{17} - 2 \beta q^{19} - 2 \beta q^{21} - 6 q^{23} + q^{25} + 4 q^{27} - 4 q^{31} + \beta q^{35} + 10 q^{37} - 2 \beta q^{41} - \beta q^{43} + q^{45} + 6 q^{47} + 5 q^{49} - 4 \beta q^{51} - 6 q^{53} + 4 \beta q^{57} - 2 \beta q^{61} + \beta q^{63} - 10 q^{67} + 12 q^{69} + 2 \beta q^{73} - 2 q^{75} - 2 \beta q^{79} - 11 q^{81} - 5 \beta q^{83} + 2 \beta q^{85} - 6 q^{89} + 8 q^{93} - 2 \beta q^{95} - 10 q^{97} +O(q^{100})$$ q - 2 * q^3 + q^5 + b * q^7 + q^9 - 2 * q^15 + 2*b * q^17 - 2*b * q^19 - 2*b * q^21 - 6 * q^23 + q^25 + 4 * q^27 - 4 * q^31 + b * q^35 + 10 * q^37 - 2*b * q^41 - b * q^43 + q^45 + 6 * q^47 + 5 * q^49 - 4*b * q^51 - 6 * q^53 + 4*b * q^57 - 2*b * q^61 + b * q^63 - 10 * q^67 + 12 * q^69 + 2*b * q^73 - 2 * q^75 - 2*b * q^79 - 11 * q^81 - 5*b * q^83 + 2*b * q^85 - 6 * q^89 + 8 * q^93 - 2*b * q^95 - 10 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 + 2 * q^5 + 2 * q^9 $$2 q - 4 q^{3} + 2 q^{5} + 2 q^{9} - 4 q^{15} - 12 q^{23} + 2 q^{25} + 8 q^{27} - 8 q^{31} + 20 q^{37} + 2 q^{45} + 12 q^{47} + 10 q^{49} - 12 q^{53} - 20 q^{67} + 24 q^{69} - 4 q^{75} - 22 q^{81} - 12 q^{89} + 16 q^{93} - 20 q^{97}+O(q^{100})$$ 2 * q - 4 * q^3 + 2 * q^5 + 2 * q^9 - 4 * q^15 - 12 * q^23 + 2 * q^25 + 8 * q^27 - 8 * q^31 + 20 * q^37 + 2 * q^45 + 12 * q^47 + 10 * q^49 - 12 * q^53 - 20 * q^67 + 24 * q^69 - 4 * q^75 - 22 * q^81 - 12 * q^89 + 16 * q^93 - 20 * 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
 −1.73205 1.73205
0 −2.00000 0 1.00000 0 −3.46410 0 1.00000 0
1.2 0 −2.00000 0 1.00000 0 3.46410 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9680.2.a.bg 2
4.b odd 2 1 605.2.a.f 2
11.b odd 2 1 inner 9680.2.a.bg 2
12.b even 2 1 5445.2.a.r 2
20.d odd 2 1 3025.2.a.j 2
44.c even 2 1 605.2.a.f 2
44.g even 10 4 605.2.g.h 8
44.h odd 10 4 605.2.g.h 8
132.d odd 2 1 5445.2.a.r 2
220.g even 2 1 3025.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.f 2 4.b odd 2 1
605.2.a.f 2 44.c even 2 1
605.2.g.h 8 44.g even 10 4
605.2.g.h 8 44.h odd 10 4
3025.2.a.j 2 20.d odd 2 1
3025.2.a.j 2 220.g even 2 1
5445.2.a.r 2 12.b even 2 1
5445.2.a.r 2 132.d odd 2 1
9680.2.a.bg 2 1.a even 1 1 trivial
9680.2.a.bg 2 11.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9680))$$:

 $$T_{3} + 2$$ T3 + 2 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{13}$$ T13 $$T_{17}^{2} - 48$$ T17^2 - 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 2)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 12$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 48$$
$19$ $$T^{2} - 48$$
$23$ $$(T + 6)^{2}$$
$29$ $$T^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$(T - 10)^{2}$$
$41$ $$T^{2} - 48$$
$43$ $$T^{2} - 12$$
$47$ $$(T - 6)^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 48$$
$67$ $$(T + 10)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 48$$
$79$ $$T^{2} - 48$$
$83$ $$T^{2} - 300$$
$89$ $$(T + 6)^{2}$$
$97$ $$(T + 10)^{2}$$