# Properties

 Label 864.2.a.l Level 864 Weight 2 Character orbit 864.a Self dual yes Analytic conductor 6.899 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 864.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{5} + 3q^{7} + O(q^{10})$$ $$q + 2q^{5} + 3q^{7} + 6q^{11} - 3q^{13} - 2q^{17} - 3q^{19} + 6q^{23} - q^{25} - 8q^{29} + 6q^{35} + 7q^{37} + 8q^{41} - 12q^{43} + 6q^{47} + 2q^{49} + 4q^{53} + 12q^{55} + 6q^{59} - q^{61} - 6q^{65} - 3q^{67} + 12q^{71} - 15q^{73} + 18q^{77} + 9q^{79} - 12q^{83} - 4q^{85} - 10q^{89} - 9q^{91} - 6q^{95} + 9q^{97} + O(q^{100})$$

## 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 0 0 2.00000 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

## 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 864.2.a.l yes 1
3.b odd 2 1 864.2.a.d yes 1
4.b odd 2 1 864.2.a.i yes 1
8.b even 2 1 1728.2.a.h 1
8.d odd 2 1 1728.2.a.e 1
9.c even 3 2 2592.2.i.c 2
9.d odd 6 2 2592.2.i.r 2
12.b even 2 1 864.2.a.a 1
24.f even 2 1 1728.2.a.u 1
24.h odd 2 1 1728.2.a.x 1
36.f odd 6 2 2592.2.i.g 2
36.h even 6 2 2592.2.i.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.a 1 12.b even 2 1
864.2.a.d yes 1 3.b odd 2 1
864.2.a.i yes 1 4.b odd 2 1
864.2.a.l yes 1 1.a even 1 1 trivial
1728.2.a.e 1 8.d odd 2 1
1728.2.a.h 1 8.b even 2 1
1728.2.a.u 1 24.f even 2 1
1728.2.a.x 1 24.h odd 2 1
2592.2.i.c 2 9.c even 3 2
2592.2.i.g 2 36.f odd 6 2
2592.2.i.r 2 9.d odd 6 2
2592.2.i.v 2 36.h even 6 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(864))$$:

 $$T_{5} - 2$$ $$T_{7} - 3$$ $$T_{11} - 6$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$1 - 2 T + 5 T^{2}$$
$7$ $$1 - 3 T + 7 T^{2}$$
$11$ $$1 - 6 T + 11 T^{2}$$
$13$ $$1 + 3 T + 13 T^{2}$$
$17$ $$1 + 2 T + 17 T^{2}$$
$19$ $$1 + 3 T + 19 T^{2}$$
$23$ $$1 - 6 T + 23 T^{2}$$
$29$ $$1 + 8 T + 29 T^{2}$$
$31$ $$1 + 31 T^{2}$$
$37$ $$1 - 7 T + 37 T^{2}$$
$41$ $$1 - 8 T + 41 T^{2}$$
$43$ $$1 + 12 T + 43 T^{2}$$
$47$ $$1 - 6 T + 47 T^{2}$$
$53$ $$1 - 4 T + 53 T^{2}$$
$59$ $$1 - 6 T + 59 T^{2}$$
$61$ $$1 + T + 61 T^{2}$$
$67$ $$1 + 3 T + 67 T^{2}$$
$71$ $$1 - 12 T + 71 T^{2}$$
$73$ $$1 + 15 T + 73 T^{2}$$
$79$ $$1 - 9 T + 79 T^{2}$$
$83$ $$1 + 12 T + 83 T^{2}$$
$89$ $$1 + 10 T + 89 T^{2}$$
$97$ $$1 - 9 T + 97 T^{2}$$
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