# Properties

 Label 576.2.s.b Level $576$ Weight $2$ Character orbit 576.s Analytic conductor $4.599$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.59938315643$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} + ( -2 + \zeta_{6} ) q^{5} + ( -1 - \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} + ( -2 + \zeta_{6} ) q^{5} + ( -1 - \zeta_{6} ) q^{7} -3 q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} -5 \zeta_{6} q^{13} -3 \zeta_{6} q^{15} + ( 4 - 8 \zeta_{6} ) q^{17} + ( 2 - 4 \zeta_{6} ) q^{19} + ( 3 - 3 \zeta_{6} ) q^{21} + 9 \zeta_{6} q^{23} + ( -2 + 2 \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -1 - \zeta_{6} ) q^{29} + ( -6 + 3 \zeta_{6} ) q^{31} + ( -3 - 3 \zeta_{6} ) q^{33} + 3 q^{35} -2 q^{37} + ( 10 - 5 \zeta_{6} ) q^{39} + ( -6 + 3 \zeta_{6} ) q^{41} + ( -3 - 3 \zeta_{6} ) q^{43} + ( 6 - 3 \zeta_{6} ) q^{45} + ( 3 - 3 \zeta_{6} ) q^{47} -4 \zeta_{6} q^{49} + 12 q^{51} + ( 3 - 6 \zeta_{6} ) q^{55} + 6 q^{57} + 3 \zeta_{6} q^{59} + ( -1 + \zeta_{6} ) q^{61} + ( 3 + 3 \zeta_{6} ) q^{63} + ( 5 + 5 \zeta_{6} ) q^{65} + ( -10 + 5 \zeta_{6} ) q^{67} + ( -18 + 9 \zeta_{6} ) q^{69} -12 q^{71} -2 q^{73} + ( -2 - 2 \zeta_{6} ) q^{75} + ( 6 - 3 \zeta_{6} ) q^{77} + ( -5 - 5 \zeta_{6} ) q^{79} + 9 q^{81} + ( -15 + 15 \zeta_{6} ) q^{83} + 12 \zeta_{6} q^{85} + ( 3 - 3 \zeta_{6} ) q^{87} + ( -4 + 8 \zeta_{6} ) q^{89} + ( -5 + 10 \zeta_{6} ) q^{91} -9 \zeta_{6} q^{93} + 6 \zeta_{6} q^{95} + ( 5 - 5 \zeta_{6} ) q^{97} + ( 9 - 9 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{5} - 3 q^{7} - 6 q^{9} + O(q^{10})$$ $$2 q - 3 q^{5} - 3 q^{7} - 6 q^{9} - 3 q^{11} - 5 q^{13} - 3 q^{15} + 3 q^{21} + 9 q^{23} - 2 q^{25} - 3 q^{29} - 9 q^{31} - 9 q^{33} + 6 q^{35} - 4 q^{37} + 15 q^{39} - 9 q^{41} - 9 q^{43} + 9 q^{45} + 3 q^{47} - 4 q^{49} + 24 q^{51} + 12 q^{57} + 3 q^{59} - q^{61} + 9 q^{63} + 15 q^{65} - 15 q^{67} - 27 q^{69} - 24 q^{71} - 4 q^{73} - 6 q^{75} + 9 q^{77} - 15 q^{79} + 18 q^{81} - 15 q^{83} + 12 q^{85} + 3 q^{87} - 9 q^{93} + 6 q^{95} + 5 q^{97} + 9 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$\zeta_{6}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 0 −1.50000 + 0.866025i 0 −1.50000 0.866025i 0 −3.00000 0
383.1 0 1.73205i 0 −1.50000 0.866025i 0 −1.50000 + 0.866025i 0 −3.00000 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
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.s.b 2
3.b odd 2 1 1728.2.s.c 2
4.b odd 2 1 576.2.s.c 2
8.b even 2 1 144.2.s.b 2
8.d odd 2 1 144.2.s.c yes 2
9.c even 3 1 1728.2.s.d 2
9.c even 3 1 5184.2.c.d 2
9.d odd 6 1 576.2.s.c 2
9.d odd 6 1 5184.2.c.b 2
12.b even 2 1 1728.2.s.d 2
24.f even 2 1 432.2.s.b 2
24.h odd 2 1 432.2.s.a 2
36.f odd 6 1 1728.2.s.c 2
36.f odd 6 1 5184.2.c.b 2
36.h even 6 1 inner 576.2.s.b 2
36.h even 6 1 5184.2.c.d 2
72.j odd 6 1 144.2.s.c yes 2
72.j odd 6 1 1296.2.c.c 2
72.l even 6 1 144.2.s.b 2
72.l even 6 1 1296.2.c.a 2
72.n even 6 1 432.2.s.b 2
72.n even 6 1 1296.2.c.a 2
72.p odd 6 1 432.2.s.a 2
72.p odd 6 1 1296.2.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.b 2 8.b even 2 1
144.2.s.b 2 72.l even 6 1
144.2.s.c yes 2 8.d odd 2 1
144.2.s.c yes 2 72.j odd 6 1
432.2.s.a 2 24.h odd 2 1
432.2.s.a 2 72.p odd 6 1
432.2.s.b 2 24.f even 2 1
432.2.s.b 2 72.n even 6 1
576.2.s.b 2 1.a even 1 1 trivial
576.2.s.b 2 36.h even 6 1 inner
576.2.s.c 2 4.b odd 2 1
576.2.s.c 2 9.d odd 6 1
1296.2.c.a 2 72.l even 6 1
1296.2.c.a 2 72.n even 6 1
1296.2.c.c 2 72.j odd 6 1
1296.2.c.c 2 72.p odd 6 1
1728.2.s.c 2 3.b odd 2 1
1728.2.s.c 2 36.f odd 6 1
1728.2.s.d 2 9.c even 3 1
1728.2.s.d 2 12.b even 2 1
5184.2.c.b 2 9.d odd 6 1
5184.2.c.b 2 36.f odd 6 1
5184.2.c.d 2 9.c even 3 1
5184.2.c.d 2 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(576, [\chi])$$:

 $$T_{5}^{2} + 3 T_{5} + 3$$ $$T_{7}^{2} + 3 T_{7} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$3 + 3 T + T^{2}$$
$7$ $$3 + 3 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$25 + 5 T + T^{2}$$
$17$ $$48 + T^{2}$$
$19$ $$12 + T^{2}$$
$23$ $$81 - 9 T + T^{2}$$
$29$ $$3 + 3 T + T^{2}$$
$31$ $$27 + 9 T + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$27 + 9 T + T^{2}$$
$43$ $$27 + 9 T + T^{2}$$
$47$ $$9 - 3 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$1 + T + T^{2}$$
$67$ $$75 + 15 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$75 + 15 T + T^{2}$$
$83$ $$225 + 15 T + T^{2}$$
$89$ $$48 + T^{2}$$
$97$ $$25 - 5 T + T^{2}$$