# Properties

 Label 576.6.a.v Level 576 Weight 6 Character orbit 576.a Self dual yes Analytic conductor 92.381 Analytic rank 0 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 32) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 14q^{5} + 208q^{7} + O(q^{10})$$ $$q + 14q^{5} + 208q^{7} + 536q^{11} - 694q^{13} + 1278q^{17} + 1112q^{19} + 3216q^{23} - 2929q^{25} + 2918q^{29} + 2624q^{31} + 2912q^{35} + 9458q^{37} - 170q^{41} - 19928q^{43} + 32q^{47} + 26457q^{49} - 22178q^{53} + 7504q^{55} - 41480q^{59} - 15462q^{61} - 9716q^{65} - 20744q^{67} + 28592q^{71} - 53670q^{73} + 111488q^{77} + 69152q^{79} + 37800q^{83} + 17892q^{85} + 126806q^{89} - 144352q^{91} + 15568q^{95} + 62290q^{97} + O(q^{100})$$

## 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 14.0000 0 208.000 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.v 1
3.b odd 2 1 64.6.a.c 1
4.b odd 2 1 576.6.a.u 1
8.b even 2 1 288.6.a.e 1
8.d odd 2 1 288.6.a.d 1
12.b even 2 1 64.6.a.e 1
24.f even 2 1 32.6.a.a 1
24.h odd 2 1 32.6.a.c yes 1
48.i odd 4 2 256.6.b.b 2
48.k even 4 2 256.6.b.h 2
120.i odd 2 1 800.6.a.a 1
120.m even 2 1 800.6.a.e 1
120.q odd 4 2 800.6.c.a 2
120.w even 4 2 800.6.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.6.a.a 1 24.f even 2 1
32.6.a.c yes 1 24.h odd 2 1
64.6.a.c 1 3.b odd 2 1
64.6.a.e 1 12.b even 2 1
256.6.b.b 2 48.i odd 4 2
256.6.b.h 2 48.k even 4 2
288.6.a.d 1 8.d odd 2 1
288.6.a.e 1 8.b even 2 1
576.6.a.u 1 4.b odd 2 1
576.6.a.v 1 1.a even 1 1 trivial
800.6.a.a 1 120.i odd 2 1
800.6.a.e 1 120.m even 2 1
800.6.c.a 2 120.q odd 4 2
800.6.c.b 2 120.w even 4 2

## Hecke kernels

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

 $$T_{5} - 14$$ $$T_{7} - 208$$ $$T_{11} - 536$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 14 T + 3125 T^{2}$$
$7$ $$1 - 208 T + 16807 T^{2}$$
$11$ $$1 - 536 T + 161051 T^{2}$$
$13$ $$1 + 694 T + 371293 T^{2}$$
$17$ $$1 - 1278 T + 1419857 T^{2}$$
$19$ $$1 - 1112 T + 2476099 T^{2}$$
$23$ $$1 - 3216 T + 6436343 T^{2}$$
$29$ $$1 - 2918 T + 20511149 T^{2}$$
$31$ $$1 - 2624 T + 28629151 T^{2}$$
$37$ $$1 - 9458 T + 69343957 T^{2}$$
$41$ $$1 + 170 T + 115856201 T^{2}$$
$43$ $$1 + 19928 T + 147008443 T^{2}$$
$47$ $$1 - 32 T + 229345007 T^{2}$$
$53$ $$1 + 22178 T + 418195493 T^{2}$$
$59$ $$1 + 41480 T + 714924299 T^{2}$$
$61$ $$1 + 15462 T + 844596301 T^{2}$$
$67$ $$1 + 20744 T + 1350125107 T^{2}$$
$71$ $$1 - 28592 T + 1804229351 T^{2}$$
$73$ $$1 + 53670 T + 2073071593 T^{2}$$
$79$ $$1 - 69152 T + 3077056399 T^{2}$$
$83$ $$1 - 37800 T + 3939040643 T^{2}$$
$89$ $$1 - 126806 T + 5584059449 T^{2}$$
$97$ $$1 - 62290 T + 8587340257 T^{2}$$