# Properties

 Label 8640.2.a.g Level $8640$ Weight $2$ Character orbit 8640.a Self dual yes Analytic conductor $68.991$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8640 = 2^{6} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8640.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$68.9907473464$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 540) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{5} - 2 q^{7}+O(q^{10})$$ q - q^5 - 2 * q^7 $$q - q^{5} - 2 q^{7} - 2 q^{13} - 3 q^{17} + 5 q^{19} - 3 q^{23} + q^{25} + 6 q^{29} - 5 q^{31} + 2 q^{35} - 2 q^{37} + 12 q^{41} + 8 q^{43} + 12 q^{47} - 3 q^{49} + 3 q^{53} + 6 q^{59} + 7 q^{61} + 2 q^{65} + 2 q^{67} - 12 q^{71} - 16 q^{73} + q^{79} - 15 q^{83} + 3 q^{85} - 12 q^{89} + 4 q^{91} - 5 q^{95} - 16 q^{97}+O(q^{100})$$ q - q^5 - 2 * q^7 - 2 * q^13 - 3 * q^17 + 5 * q^19 - 3 * q^23 + q^25 + 6 * q^29 - 5 * q^31 + 2 * q^35 - 2 * q^37 + 12 * q^41 + 8 * q^43 + 12 * q^47 - 3 * q^49 + 3 * q^53 + 6 * q^59 + 7 * q^61 + 2 * q^65 + 2 * q^67 - 12 * q^71 - 16 * q^73 + q^79 - 15 * q^83 + 3 * q^85 - 12 * q^89 + 4 * q^91 - 5 * q^95 - 16 * 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 −2.00000 0 0 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 8640.2.a.g 1
3.b odd 2 1 8640.2.a.bl 1
4.b odd 2 1 8640.2.a.w 1
8.b even 2 1 2160.2.a.o 1
8.d odd 2 1 540.2.a.f yes 1
12.b even 2 1 8640.2.a.bz 1
24.f even 2 1 540.2.a.c 1
24.h odd 2 1 2160.2.a.c 1
40.e odd 2 1 2700.2.a.g 1
40.k even 4 2 2700.2.d.f 2
72.l even 6 2 1620.2.i.i 2
72.p odd 6 2 1620.2.i.a 2
120.m even 2 1 2700.2.a.f 1
120.q odd 4 2 2700.2.d.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.a.c 1 24.f even 2 1
540.2.a.f yes 1 8.d odd 2 1
1620.2.i.a 2 72.p odd 6 2
1620.2.i.i 2 72.l even 6 2
2160.2.a.c 1 24.h odd 2 1
2160.2.a.o 1 8.b even 2 1
2700.2.a.f 1 120.m even 2 1
2700.2.a.g 1 40.e odd 2 1
2700.2.d.e 2 120.q odd 4 2
2700.2.d.f 2 40.k even 4 2
8640.2.a.g 1 1.a even 1 1 trivial
8640.2.a.w 1 4.b odd 2 1
8640.2.a.bl 1 3.b odd 2 1
8640.2.a.bz 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8640))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{11}$$ T11 $$T_{13} + 2$$ T13 + 2 $$T_{17} + 3$$ T17 + 3 $$T_{19} - 5$$ T19 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T + 2$$
$11$ $$T$$
$13$ $$T + 2$$
$17$ $$T + 3$$
$19$ $$T - 5$$
$23$ $$T + 3$$
$29$ $$T - 6$$
$31$ $$T + 5$$
$37$ $$T + 2$$
$41$ $$T - 12$$
$43$ $$T - 8$$
$47$ $$T - 12$$
$53$ $$T - 3$$
$59$ $$T - 6$$
$61$ $$T - 7$$
$67$ $$T - 2$$
$71$ $$T + 12$$
$73$ $$T + 16$$
$79$ $$T - 1$$
$83$ $$T + 15$$
$89$ $$T + 12$$
$97$ $$T + 16$$