# Properties

 Label 576.7.e.b Level $576$ Weight $7$ Character orbit 576.e Analytic conductor $132.511$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 576.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$132.511152165$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 41 \beta q^{5} - 484 q^{7}+O(q^{10})$$ q + 41*b * q^5 - 484 * q^7 $$q + 41 \beta q^{5} - 484 q^{7} + 316 \beta q^{11} - 3368 q^{13} - 3 \beta q^{17} - 5744 q^{19} + 796 \beta q^{23} - 14633 q^{25} + 6919 \beta q^{29} - 39796 q^{31} - 19844 \beta q^{35} - 52526 q^{37} + 8731 \beta q^{41} - 3800 q^{43} - 18100 \beta q^{47} + 116607 q^{49} + 56271 \beta q^{53} - 233208 q^{55} - 58888 \beta q^{59} - 13250 q^{61} - 138088 \beta q^{65} - 168968 q^{67} + 125268 \beta q^{71} + 236144 q^{73} - 152944 \beta q^{77} - 35116 q^{79} - 2588 \beta q^{83} + 2214 q^{85} + 30483 \beta q^{89} + 1630112 q^{91} - 235504 \beta q^{95} - 321424 q^{97} +O(q^{100})$$ q + 41*b * q^5 - 484 * q^7 + 316*b * q^11 - 3368 * q^13 - 3*b * q^17 - 5744 * q^19 + 796*b * q^23 - 14633 * q^25 + 6919*b * q^29 - 39796 * q^31 - 19844*b * q^35 - 52526 * q^37 + 8731*b * q^41 - 3800 * q^43 - 18100*b * q^47 + 116607 * q^49 + 56271*b * q^53 - 233208 * q^55 - 58888*b * q^59 - 13250 * q^61 - 138088*b * q^65 - 168968 * q^67 + 125268*b * q^71 + 236144 * q^73 - 152944*b * q^77 - 35116 * q^79 - 2588*b * q^83 + 2214 * q^85 + 30483*b * q^89 + 1630112 * q^91 - 235504*b * q^95 - 321424 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 968 q^{7}+O(q^{10})$$ 2 * q - 968 * q^7 $$2 q - 968 q^{7} - 6736 q^{13} - 11488 q^{19} - 29266 q^{25} - 79592 q^{31} - 105052 q^{37} - 7600 q^{43} + 233214 q^{49} - 466416 q^{55} - 26500 q^{61} - 337936 q^{67} + 472288 q^{73} - 70232 q^{79} + 4428 q^{85} + 3260224 q^{91} - 642848 q^{97}+O(q^{100})$$ 2 * q - 968 * q^7 - 6736 * q^13 - 11488 * q^19 - 29266 * q^25 - 79592 * q^31 - 105052 * q^37 - 7600 * q^43 + 233214 * q^49 - 466416 * q^55 - 26500 * q^61 - 337936 * q^67 + 472288 * q^73 - 70232 * q^79 + 4428 * q^85 + 3260224 * q^91 - 642848 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.41421i 1.41421i
0 0 0 173.948i 0 −484.000 0 0 0
449.2 0 0 0 173.948i 0 −484.000 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.7.e.b 2
3.b odd 2 1 inner 576.7.e.b 2
4.b odd 2 1 576.7.e.k 2
8.b even 2 1 18.7.b.a 2
8.d odd 2 1 144.7.e.d 2
12.b even 2 1 576.7.e.k 2
24.f even 2 1 144.7.e.d 2
24.h odd 2 1 18.7.b.a 2
40.f even 2 1 450.7.d.a 2
40.i odd 4 2 450.7.b.a 4
72.j odd 6 2 162.7.d.d 4
72.n even 6 2 162.7.d.d 4
120.i odd 2 1 450.7.d.a 2
120.w even 4 2 450.7.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.b.a 2 8.b even 2 1
18.7.b.a 2 24.h odd 2 1
144.7.e.d 2 8.d odd 2 1
144.7.e.d 2 24.f even 2 1
162.7.d.d 4 72.j odd 6 2
162.7.d.d 4 72.n even 6 2
450.7.b.a 4 40.i odd 4 2
450.7.b.a 4 120.w even 4 2
450.7.d.a 2 40.f even 2 1
450.7.d.a 2 120.i odd 2 1
576.7.e.b 2 1.a even 1 1 trivial
576.7.e.b 2 3.b odd 2 1 inner
576.7.e.k 2 4.b odd 2 1
576.7.e.k 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{7}^{\mathrm{new}}(576, [\chi])$$:

 $$T_{5}^{2} + 30258$$ T5^2 + 30258 $$T_{7} + 484$$ T7 + 484

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 30258$$
$7$ $$(T + 484)^{2}$$
$11$ $$T^{2} + 1797408$$
$13$ $$(T + 3368)^{2}$$
$17$ $$T^{2} + 162$$
$19$ $$(T + 5744)^{2}$$
$23$ $$T^{2} + 11405088$$
$29$ $$T^{2} + 861706098$$
$31$ $$(T + 39796)^{2}$$
$37$ $$(T + 52526)^{2}$$
$41$ $$T^{2} + 1372146498$$
$43$ $$(T + 3800)^{2}$$
$47$ $$T^{2} + 5896980000$$
$53$ $$T^{2} + 56995657938$$
$59$ $$T^{2} + 62420337792$$
$61$ $$(T + 13250)^{2}$$
$67$ $$(T + 168968)^{2}$$
$71$ $$T^{2} + 282457292832$$
$73$ $$(T - 236144)^{2}$$
$79$ $$(T + 35116)^{2}$$
$83$ $$T^{2} + 120559392$$
$89$ $$T^{2} + 16725839202$$
$97$ $$(T + 321424)^{2}$$