# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{5} -\beta q^{7} +O(q^{10})$$ $$q -\beta q^{5} -\beta q^{7} + q^{11} + 4 q^{13} + 2 \beta q^{19} -6 q^{23} + 8 q^{25} + 2 \beta q^{29} + \beta q^{31} + 13 q^{35} + 10 q^{37} + 2 \beta q^{41} -2 \beta q^{43} -10 q^{47} + 6 q^{49} + \beta q^{53} -\beta q^{55} -4 q^{59} -4 \beta q^{65} -2 \beta q^{67} + 8 q^{71} -3 q^{73} -\beta q^{77} + 4 \beta q^{79} + 9 q^{83} -2 \beta q^{89} -4 \beta q^{91} -26 q^{95} + 7 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 2q^{11} + 8q^{13} - 12q^{23} + 16q^{25} + 26q^{35} + 20q^{37} - 20q^{47} + 12q^{49} - 8q^{59} + 16q^{71} - 6q^{73} + 18q^{83} - 52q^{95} + 14q^{97} + O(q^{100})$$

## 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
 2.30278 −1.30278
0 0 0 −3.60555 0 −3.60555 0 0 0
1.2 0 0 0 3.60555 0 3.60555 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.a.n yes 2
3.b odd 2 1 864.2.a.m 2
4.b odd 2 1 864.2.a.m 2
8.b even 2 1 1728.2.a.bc 2
8.d odd 2 1 1728.2.a.bd 2
9.c even 3 2 2592.2.i.bb 4
9.d odd 6 2 2592.2.i.bc 4
12.b even 2 1 inner 864.2.a.n yes 2
24.f even 2 1 1728.2.a.bc 2
24.h odd 2 1 1728.2.a.bd 2
36.f odd 6 2 2592.2.i.bc 4
36.h even 6 2 2592.2.i.bb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.m 2 3.b odd 2 1
864.2.a.m 2 4.b odd 2 1
864.2.a.n yes 2 1.a even 1 1 trivial
864.2.a.n yes 2 12.b even 2 1 inner
1728.2.a.bc 2 8.b even 2 1
1728.2.a.bc 2 24.f even 2 1
1728.2.a.bd 2 8.d odd 2 1
1728.2.a.bd 2 24.h odd 2 1
2592.2.i.bb 4 9.c even 3 2
2592.2.i.bb 4 36.h even 6 2
2592.2.i.bc 4 9.d odd 6 2
2592.2.i.bc 4 36.f odd 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(864))$$:

 $$T_{5}^{2} - 13$$ $$T_{7}^{2} - 13$$ $$T_{11} - 1$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$1 - 3 T^{2} + 25 T^{4}$$
$7$ $$1 + T^{2} + 49 T^{4}$$
$11$ $$( 1 - T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 4 T + 13 T^{2} )^{2}$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$1 - 14 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 6 T + 23 T^{2} )^{2}$$
$29$ $$1 + 6 T^{2} + 841 T^{4}$$
$31$ $$1 + 49 T^{2} + 961 T^{4}$$
$37$ $$( 1 - 10 T + 37 T^{2} )^{2}$$
$41$ $$1 + 30 T^{2} + 1681 T^{4}$$
$43$ $$1 + 34 T^{2} + 1849 T^{4}$$
$47$ $$( 1 + 10 T + 47 T^{2} )^{2}$$
$53$ $$1 + 93 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 4 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 61 T^{2} )^{2}$$
$67$ $$1 + 82 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 8 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 3 T + 73 T^{2} )^{2}$$
$79$ $$1 - 50 T^{2} + 6241 T^{4}$$
$83$ $$( 1 - 9 T + 83 T^{2} )^{2}$$
$89$ $$1 + 126 T^{2} + 7921 T^{4}$$
$97$ $$( 1 - 7 T + 97 T^{2} )^{2}$$