# Properties

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

# Related objects

## 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

 $$f(q)$$ $$=$$ $$q - q^{5} + 3 q^{7} + O(q^{10})$$ $$q - q^{5} + 3 q^{7} - 3 q^{11} + 4 q^{17} + 6 q^{19} + 6 q^{23} - 4 q^{25} - 2 q^{29} + 9 q^{31} - 3 q^{35} - 2 q^{37} - 10 q^{41} + 6 q^{43} + 6 q^{47} + 2 q^{49} + 13 q^{53} + 3 q^{55} - 12 q^{59} + 8 q^{61} + 6 q^{67} + 12 q^{71} + 9 q^{73} - 9 q^{77} - 3 q^{83} - 4 q^{85} + 14 q^{89} - 6 q^{95} - 9 q^{97} + O(q^{100})$$

## 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 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
$$2$$ $$1$$
$$3$$ $$-1$$

## 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.f yes 1
3.b odd 2 1 864.2.a.h yes 1
4.b odd 2 1 864.2.a.e 1
8.b even 2 1 1728.2.a.t 1
8.d odd 2 1 1728.2.a.q 1
9.c even 3 2 2592.2.i.m 2
9.d odd 6 2 2592.2.i.i 2
12.b even 2 1 864.2.a.g yes 1
24.f even 2 1 1728.2.a.j 1
24.h odd 2 1 1728.2.a.k 1
36.f odd 6 2 2592.2.i.p 2
36.h even 6 2 2592.2.i.l 2

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

## Hecke kernels

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

 $$T_{5} + 1$$ $$T_{7} - 3$$ $$T_{11} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$-3 + T$$
$11$ $$3 + T$$
$13$ $$T$$
$17$ $$-4 + T$$
$19$ $$-6 + T$$
$23$ $$-6 + T$$
$29$ $$2 + T$$
$31$ $$-9 + T$$
$37$ $$2 + T$$
$41$ $$10 + T$$
$43$ $$-6 + T$$
$47$ $$-6 + T$$
$53$ $$-13 + T$$
$59$ $$12 + T$$
$61$ $$-8 + T$$
$67$ $$-6 + T$$
$71$ $$-12 + T$$
$73$ $$-9 + T$$
$79$ $$T$$
$83$ $$3 + T$$
$89$ $$-14 + T$$
$97$ $$9 + T$$