# Properties

 Label 576.6.a.h Level $576$ Weight $6$ Character orbit 576.a Self dual yes Analytic conductor $92.381$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 576.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$92.3810802123$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 8) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 74 q^{5} + 24 q^{7}+O(q^{10})$$ q - 74 * q^5 + 24 * q^7 $$q - 74 q^{5} + 24 q^{7} - 124 q^{11} - 478 q^{13} + 1198 q^{17} + 3044 q^{19} + 184 q^{23} + 2351 q^{25} - 3282 q^{29} + 5728 q^{31} - 1776 q^{35} - 10326 q^{37} + 8886 q^{41} - 9188 q^{43} + 23664 q^{47} - 16231 q^{49} + 11686 q^{53} + 9176 q^{55} - 16876 q^{59} + 18482 q^{61} + 35372 q^{65} - 15532 q^{67} - 31960 q^{71} - 4886 q^{73} - 2976 q^{77} - 44560 q^{79} - 67364 q^{83} - 88652 q^{85} - 71994 q^{89} - 11472 q^{91} - 225256 q^{95} + 48866 q^{97}+O(q^{100})$$ q - 74 * q^5 + 24 * q^7 - 124 * q^11 - 478 * q^13 + 1198 * q^17 + 3044 * q^19 + 184 * q^23 + 2351 * q^25 - 3282 * q^29 + 5728 * q^31 - 1776 * q^35 - 10326 * q^37 + 8886 * q^41 - 9188 * q^43 + 23664 * q^47 - 16231 * q^49 + 11686 * q^53 + 9176 * q^55 - 16876 * q^59 + 18482 * q^61 + 35372 * q^65 - 15532 * q^67 - 31960 * q^71 - 4886 * q^73 - 2976 * q^77 - 44560 * q^79 - 67364 * q^83 - 88652 * q^85 - 71994 * q^89 - 11472 * q^91 - 225256 * q^95 + 48866 * 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 −74.0000 0 24.0000 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 576.6.a.h 1
3.b odd 2 1 64.6.a.g 1
4.b odd 2 1 576.6.a.g 1
8.b even 2 1 144.6.a.k 1
8.d odd 2 1 72.6.a.f 1
12.b even 2 1 64.6.a.a 1
24.f even 2 1 8.6.a.a 1
24.h odd 2 1 16.6.a.a 1
48.i odd 4 2 256.6.b.d 2
48.k even 4 2 256.6.b.f 2
120.i odd 2 1 400.6.a.l 1
120.m even 2 1 200.6.a.a 1
120.q odd 4 2 200.6.c.a 2
120.w even 4 2 400.6.c.d 2
168.e odd 2 1 392.6.a.b 1
168.i even 2 1 784.6.a.l 1
168.v even 6 2 392.6.i.b 2
168.be odd 6 2 392.6.i.e 2
264.p odd 2 1 968.6.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.a.a 1 24.f even 2 1
16.6.a.a 1 24.h odd 2 1
64.6.a.a 1 12.b even 2 1
64.6.a.g 1 3.b odd 2 1
72.6.a.f 1 8.d odd 2 1
144.6.a.k 1 8.b even 2 1
200.6.a.a 1 120.m even 2 1
200.6.c.a 2 120.q odd 4 2
256.6.b.d 2 48.i odd 4 2
256.6.b.f 2 48.k even 4 2
392.6.a.b 1 168.e odd 2 1
392.6.i.b 2 168.v even 6 2
392.6.i.e 2 168.be odd 6 2
400.6.a.l 1 120.i odd 2 1
400.6.c.d 2 120.w even 4 2
576.6.a.g 1 4.b odd 2 1
576.6.a.h 1 1.a even 1 1 trivial
784.6.a.l 1 168.i even 2 1
968.6.a.a 1 264.p odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(576))$$:

 $$T_{5} + 74$$ T5 + 74 $$T_{7} - 24$$ T7 - 24 $$T_{11} + 124$$ T11 + 124

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 74$$
$7$ $$T - 24$$
$11$ $$T + 124$$
$13$ $$T + 478$$
$17$ $$T - 1198$$
$19$ $$T - 3044$$
$23$ $$T - 184$$
$29$ $$T + 3282$$
$31$ $$T - 5728$$
$37$ $$T + 10326$$
$41$ $$T - 8886$$
$43$ $$T + 9188$$
$47$ $$T - 23664$$
$53$ $$T - 11686$$
$59$ $$T + 16876$$
$61$ $$T - 18482$$
$67$ $$T + 15532$$
$71$ $$T + 31960$$
$73$ $$T + 4886$$
$79$ $$T + 44560$$
$83$ $$T + 67364$$
$89$ $$T + 71994$$
$97$ $$T - 48866$$