# Properties

 Label 576.2.r.c Level $576$ Weight $2$ Character orbit 576.r Analytic conductor $4.599$ Analytic rank $0$ Dimension $8$ CM discriminant -8 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 576.r (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.59938315643$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} - \beta_{4}) q^{3} + ( - \beta_{7} - \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10})$$ q + (b6 - b4) * q^3 + (-b7 - b2 + b1 - 1) * q^9 $$q + (\beta_{6} - \beta_{4}) q^{3} + ( - \beta_{7} - \beta_{2} + \beta_1 - 1) q^{9} + (2 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3}) q^{11} + ( - 2 \beta_{7} - \beta_{2} + 3) q^{17} + (3 \beta_{6} - \beta_{5} - 2 \beta_{4} + 3 \beta_{3}) q^{19} - 5 \beta_1 q^{25} + ( - 3 \beta_{5} + 2 \beta_{4}) q^{27} + ( - \beta_{7} - 2 \beta_{2} - 5 \beta_1 + 1) q^{33} + ( - 2 \beta_{7} + 2 \beta_{2} + 3 \beta_1 - 3) q^{41} + ( - \beta_{6} - \beta_{5} + 4 \beta_{4} + 4 \beta_{3}) q^{43} + ( - 7 \beta_1 + 7) q^{49} + (4 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3}) q^{51} + ( - 2 \beta_{7} - 3 \beta_{2} + 5 \beta_1 - 12) q^{57} + ( - 4 \beta_{6} + 5 \beta_{5} + 5 \beta_{4} - \beta_{3}) q^{59} + ( - 2 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 5 \beta_{3}) q^{67} + (6 \beta_{7} + 3 \beta_{2} - 1) q^{73} - 5 \beta_{6} q^{75} + (2 \beta_{7} + 7 \beta_1) q^{81} + (9 \beta_{6} + 9 \beta_{5} - 9 \beta_{4} - 9 \beta_{3}) q^{83} + 18 q^{89} + (3 \beta_{7} + 6 \beta_{2} + 5 \beta_1) q^{97} + ( - 5 \beta_{6} - 6 \beta_{5} + \beta_{4} + 3 \beta_{3}) q^{99}+O(q^{100})$$ q + (b6 - b4) * q^3 + (-b7 - b2 + b1 - 1) * q^9 + (2*b6 + 2*b5 - b4 - b3) * q^11 + (-2*b7 - b2 + 3) * q^17 + (3*b6 - b5 - 2*b4 + 3*b3) * q^19 - 5*b1 * q^25 + (-3*b5 + 2*b4) * q^27 + (-b7 - 2*b2 - 5*b1 + 1) * q^33 + (-2*b7 + 2*b2 + 3*b1 - 3) * q^41 + (-b6 - b5 + 4*b4 + 4*b3) * q^43 + (-7*b1 + 7) * q^49 + (4*b6 - 3*b5 - 2*b4 - 3*b3) * q^51 + (-2*b7 - 3*b2 + 5*b1 - 12) * q^57 + (-4*b6 + 5*b5 + 5*b4 - b3) * q^59 + (-2*b6 - 3*b5 - 3*b4 + 5*b3) * q^67 + (6*b7 + 3*b2 - 1) * q^73 - 5*b6 * q^75 + (2*b7 + 7*b1) * q^81 + (9*b6 + 9*b5 - 9*b4 - 9*b3) * q^83 + 18 * q^89 + (3*b7 + 6*b2 + 5*b1) * q^97 + (-5*b6 - 6*b5 + b4 + 3*b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{9}+O(q^{10})$$ 8 * q - 4 * q^9 $$8 q - 4 q^{9} + 24 q^{17} - 20 q^{25} - 12 q^{33} - 12 q^{41} + 28 q^{49} - 76 q^{57} - 8 q^{73} + 28 q^{81} + 144 q^{89} + 20 q^{97}+O(q^{100})$$ 8 * q - 4 * q^9 + 24 * q^17 - 20 * q^25 - 12 * q^33 - 12 * q^41 + 28 * q^49 - 76 * q^57 - 8 * q^73 + 28 * q^81 + 144 * q^89 + 20 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{2}$$ $$=$$ $$2\zeta_{24}^{5} + 2\zeta_{24}^{3} - 2\zeta_{24}$$ 2*v^5 + 2*v^3 - 2*v $$\beta_{3}$$ $$=$$ $$-\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}^{2}$$ -v^7 - v^5 + v^3 - v^2 $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ v^6 - v^5 + v^3 + v $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -v^6 - v^5 + v^3 + v $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}^{2}$$ -v^7 - v^5 + v^3 + v^2 $$\beta_{7}$$ $$=$$ $$-2\zeta_{24}^{7} + 2\zeta_{24}$$ -2*v^7 + 2*v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} ) / 4$$ (b7 - b6 + b5 + b4 - b3) / 4 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{6} - \beta_{3} ) / 2$$ (b6 - b3) / 2 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{5} + \beta_{4} + \beta_{2} ) / 4$$ (b5 + b4 + b2) / 4 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{5}$$ $$=$$ $$( \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2} ) / 4$$ (b7 - b6 - b3 + b2) / 4 $$\zeta_{24}^{6}$$ $$=$$ $$( -\beta_{5} + \beta_{4} ) / 2$$ (-b5 + b4) / 2 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} ) / 4$$ (-b7 - b6 + b5 + b4 - b3) / 4

## Character values

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

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$-1 + \beta_{1}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 −0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i
0 −1.57313 + 0.724745i 0 0 0 0 0 1.94949 2.28024i 0
97.2 0 −0.158919 1.72474i 0 0 0 0 0 −2.94949 + 0.548188i 0
97.3 0 0.158919 + 1.72474i 0 0 0 0 0 −2.94949 + 0.548188i 0
97.4 0 1.57313 0.724745i 0 0 0 0 0 1.94949 2.28024i 0
481.1 0 −1.57313 0.724745i 0 0 0 0 0 1.94949 + 2.28024i 0
481.2 0 −0.158919 + 1.72474i 0 0 0 0 0 −2.94949 0.548188i 0
481.3 0 0.158919 1.72474i 0 0 0 0 0 −2.94949 0.548188i 0
481.4 0 1.57313 + 0.724745i 0 0 0 0 0 1.94949 + 2.28024i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 481.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b even 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
72.n even 6 1 inner
72.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.r.c 8
3.b odd 2 1 1728.2.r.c 8
4.b odd 2 1 inner 576.2.r.c 8
8.b even 2 1 inner 576.2.r.c 8
8.d odd 2 1 CM 576.2.r.c 8
9.c even 3 1 inner 576.2.r.c 8
9.c even 3 1 5184.2.d.l 4
9.d odd 6 1 1728.2.r.c 8
9.d odd 6 1 5184.2.d.e 4
12.b even 2 1 1728.2.r.c 8
24.f even 2 1 1728.2.r.c 8
24.h odd 2 1 1728.2.r.c 8
36.f odd 6 1 inner 576.2.r.c 8
36.f odd 6 1 5184.2.d.l 4
36.h even 6 1 1728.2.r.c 8
36.h even 6 1 5184.2.d.e 4
72.j odd 6 1 1728.2.r.c 8
72.j odd 6 1 5184.2.d.e 4
72.l even 6 1 1728.2.r.c 8
72.l even 6 1 5184.2.d.e 4
72.n even 6 1 inner 576.2.r.c 8
72.n even 6 1 5184.2.d.l 4
72.p odd 6 1 inner 576.2.r.c 8
72.p odd 6 1 5184.2.d.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.c 8 1.a even 1 1 trivial
576.2.r.c 8 4.b odd 2 1 inner
576.2.r.c 8 8.b even 2 1 inner
576.2.r.c 8 8.d odd 2 1 CM
576.2.r.c 8 9.c even 3 1 inner
576.2.r.c 8 36.f odd 6 1 inner
576.2.r.c 8 72.n even 6 1 inner
576.2.r.c 8 72.p odd 6 1 inner
1728.2.r.c 8 3.b odd 2 1
1728.2.r.c 8 9.d odd 6 1
1728.2.r.c 8 12.b even 2 1
1728.2.r.c 8 24.f even 2 1
1728.2.r.c 8 24.h odd 2 1
1728.2.r.c 8 36.h even 6 1
1728.2.r.c 8 72.j odd 6 1
1728.2.r.c 8 72.l even 6 1
5184.2.d.e 4 9.d odd 6 1
5184.2.d.e 4 36.h even 6 1
5184.2.d.e 4 72.j odd 6 1
5184.2.d.e 4 72.l even 6 1
5184.2.d.l 4 9.c even 3 1
5184.2.d.l 4 36.f odd 6 1
5184.2.d.l 4 72.n even 6 1
5184.2.d.l 4 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{2}^{\mathrm{new}}(576, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + 2 T^{6} - 5 T^{4} + 18 T^{2} + \cdots + 81$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$T^{8} - 30 T^{6} + 891 T^{4} + \cdots + 81$$
$13$ $$T^{8}$$
$17$ $$(T^{2} - 6 T - 15)^{4}$$
$19$ $$(T^{4} + 110 T^{2} + 2809)^{2}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$(T^{4} + 6 T^{3} + 123 T^{2} - 522 T + 7569)^{2}$$
$43$ $$T^{8} - 158 T^{6} + 24123 T^{4} + \cdots + 707281$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$T^{8} - 318 T^{6} + \cdots + 395254161$$
$61$ $$T^{8}$$
$67$ $$T^{8} - 206 T^{6} + 42411 T^{4} + \cdots + 625$$
$71$ $$T^{8}$$
$73$ $$(T^{2} + 2 T - 215)^{4}$$
$79$ $$T^{8}$$
$83$ $$(T^{4} - 324 T^{2} + 104976)^{2}$$
$89$ $$(T - 18)^{8}$$
$97$ $$(T^{4} - 10 T^{3} + 291 T^{2} + \cdots + 36481)^{2}$$