# Properties

 Label 8640.2.a.bb Level $8640$ Weight $2$ Character orbit 8640.a Self dual yes Analytic conductor $68.991$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 135) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{5} + 3 q^{7}+O(q^{10})$$ q - q^5 + 3 * q^7 $$q - q^{5} + 3 q^{7} + 2 q^{11} + 5 q^{13} + 8 q^{17} + q^{19} + 6 q^{23} + q^{25} + 2 q^{29} - 3 q^{35} - 5 q^{37} + 10 q^{41} + 4 q^{43} + 4 q^{47} + 2 q^{49} - 2 q^{53} - 2 q^{55} + 8 q^{59} - 7 q^{61} - 5 q^{65} - 9 q^{67} + 2 q^{71} - 5 q^{73} + 6 q^{77} + 3 q^{79} - 6 q^{83} - 8 q^{85} + 12 q^{89} + 15 q^{91} - q^{95} - 13 q^{97}+O(q^{100})$$ q - q^5 + 3 * q^7 + 2 * q^11 + 5 * q^13 + 8 * q^17 + q^19 + 6 * q^23 + q^25 + 2 * q^29 - 3 * q^35 - 5 * q^37 + 10 * q^41 + 4 * q^43 + 4 * q^47 + 2 * q^49 - 2 * q^53 - 2 * q^55 + 8 * q^59 - 7 * q^61 - 5 * q^65 - 9 * q^67 + 2 * q^71 - 5 * q^73 + 6 * q^77 + 3 * q^79 - 6 * q^83 - 8 * q^85 + 12 * q^89 + 15 * q^91 - q^95 - 13 * 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 3.00000 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.bb 1
3.b odd 2 1 8640.2.a.ce 1
4.b odd 2 1 8640.2.a.c 1
8.b even 2 1 2160.2.a.v 1
8.d odd 2 1 135.2.a.b yes 1
12.b even 2 1 8640.2.a.bh 1
24.f even 2 1 135.2.a.a 1
24.h odd 2 1 2160.2.a.j 1
40.e odd 2 1 675.2.a.a 1
40.k even 4 2 675.2.b.b 2
56.e even 2 1 6615.2.a.j 1
72.l even 6 2 405.2.e.h 2
72.p odd 6 2 405.2.e.b 2
120.m even 2 1 675.2.a.i 1
120.q odd 4 2 675.2.b.a 2
168.e odd 2 1 6615.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.a.a 1 24.f even 2 1
135.2.a.b yes 1 8.d odd 2 1
405.2.e.b 2 72.p odd 6 2
405.2.e.h 2 72.l even 6 2
675.2.a.a 1 40.e odd 2 1
675.2.a.i 1 120.m even 2 1
675.2.b.a 2 120.q odd 4 2
675.2.b.b 2 40.k even 4 2
2160.2.a.j 1 24.h odd 2 1
2160.2.a.v 1 8.b even 2 1
6615.2.a.a 1 168.e odd 2 1
6615.2.a.j 1 56.e even 2 1
8640.2.a.c 1 4.b odd 2 1
8640.2.a.bb 1 1.a even 1 1 trivial
8640.2.a.bh 1 12.b even 2 1
8640.2.a.ce 1 3.b odd 2 1

## Hecke kernels

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

 $$T_{7} - 3$$ T7 - 3 $$T_{11} - 2$$ T11 - 2 $$T_{13} - 5$$ T13 - 5 $$T_{17} - 8$$ T17 - 8 $$T_{19} - 1$$ T19 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T - 3$$
$11$ $$T - 2$$
$13$ $$T - 5$$
$17$ $$T - 8$$
$19$ $$T - 1$$
$23$ $$T - 6$$
$29$ $$T - 2$$
$31$ $$T$$
$37$ $$T + 5$$
$41$ $$T - 10$$
$43$ $$T - 4$$
$47$ $$T - 4$$
$53$ $$T + 2$$
$59$ $$T - 8$$
$61$ $$T + 7$$
$67$ $$T + 9$$
$71$ $$T - 2$$
$73$ $$T + 5$$
$79$ $$T - 3$$
$83$ $$T + 6$$
$89$ $$T - 12$$
$97$ $$T + 13$$