# Properties

 Label 576.3.g.e Level $576$ Weight $3$ Character orbit 576.g Analytic conductor $15.695$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$15.6948632272$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 q^{5} + ( 4 - 8 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -2 q^{5} + ( 4 - 8 \zeta_{6} ) q^{7} + ( -4 + 8 \zeta_{6} ) q^{11} -2 q^{13} -10 q^{17} + ( 12 - 24 \zeta_{6} ) q^{19} + ( -16 + 32 \zeta_{6} ) q^{23} -21 q^{25} -26 q^{29} + ( -4 + 8 \zeta_{6} ) q^{31} + ( -8 + 16 \zeta_{6} ) q^{35} -26 q^{37} -58 q^{41} + ( -28 + 56 \zeta_{6} ) q^{43} + ( 40 - 80 \zeta_{6} ) q^{47} + q^{49} -74 q^{53} + ( 8 - 16 \zeta_{6} ) q^{55} + ( 52 - 104 \zeta_{6} ) q^{59} -26 q^{61} + 4 q^{65} + ( 4 - 8 \zeta_{6} ) q^{67} -46 q^{73} + 48 q^{77} + ( -68 + 136 \zeta_{6} ) q^{79} + ( -28 + 56 \zeta_{6} ) q^{83} + 20 q^{85} -82 q^{89} + ( -8 + 16 \zeta_{6} ) q^{91} + ( -24 + 48 \zeta_{6} ) q^{95} + 2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} + O(q^{10})$$ $$2q - 4q^{5} - 4q^{13} - 20q^{17} - 42q^{25} - 52q^{29} - 52q^{37} - 116q^{41} + 2q^{49} - 148q^{53} - 52q^{61} + 8q^{65} - 92q^{73} + 96q^{77} + 40q^{85} - 164q^{89} + 4q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −2.00000 0 6.92820i 0 0 0
127.2 0 0 0 −2.00000 0 6.92820i 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.3.g.e 2
3.b odd 2 1 192.3.g.b 2
4.b odd 2 1 inner 576.3.g.e 2
8.b even 2 1 36.3.d.c 2
8.d odd 2 1 36.3.d.c 2
12.b even 2 1 192.3.g.b 2
16.e even 4 2 2304.3.b.l 4
16.f odd 4 2 2304.3.b.l 4
24.f even 2 1 12.3.d.a 2
24.h odd 2 1 12.3.d.a 2
40.e odd 2 1 900.3.c.e 2
40.f even 2 1 900.3.c.e 2
40.i odd 4 2 900.3.f.c 4
40.k even 4 2 900.3.f.c 4
48.i odd 4 2 768.3.b.c 4
48.k even 4 2 768.3.b.c 4
72.j odd 6 1 324.3.f.d 2
72.j odd 6 1 324.3.f.j 2
72.l even 6 1 324.3.f.d 2
72.l even 6 1 324.3.f.j 2
72.n even 6 1 324.3.f.a 2
72.n even 6 1 324.3.f.g 2
72.p odd 6 1 324.3.f.a 2
72.p odd 6 1 324.3.f.g 2
120.i odd 2 1 300.3.c.b 2
120.m even 2 1 300.3.c.b 2
120.q odd 4 2 300.3.f.a 4
120.w even 4 2 300.3.f.a 4
168.e odd 2 1 588.3.g.b 2
168.i even 2 1 588.3.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.d.a 2 24.f even 2 1
12.3.d.a 2 24.h odd 2 1
36.3.d.c 2 8.b even 2 1
36.3.d.c 2 8.d odd 2 1
192.3.g.b 2 3.b odd 2 1
192.3.g.b 2 12.b even 2 1
300.3.c.b 2 120.i odd 2 1
300.3.c.b 2 120.m even 2 1
300.3.f.a 4 120.q odd 4 2
300.3.f.a 4 120.w even 4 2
324.3.f.a 2 72.n even 6 1
324.3.f.a 2 72.p odd 6 1
324.3.f.d 2 72.j odd 6 1
324.3.f.d 2 72.l even 6 1
324.3.f.g 2 72.n even 6 1
324.3.f.g 2 72.p odd 6 1
324.3.f.j 2 72.j odd 6 1
324.3.f.j 2 72.l even 6 1
576.3.g.e 2 1.a even 1 1 trivial
576.3.g.e 2 4.b odd 2 1 inner
588.3.g.b 2 168.e odd 2 1
588.3.g.b 2 168.i even 2 1
768.3.b.c 4 48.i odd 4 2
768.3.b.c 4 48.k even 4 2
900.3.c.e 2 40.e odd 2 1
900.3.c.e 2 40.f even 2 1
900.3.f.c 4 40.i odd 4 2
900.3.f.c 4 40.k even 4 2
2304.3.b.l 4 16.e even 4 2
2304.3.b.l 4 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 2$$ acting on $$S_{3}^{\mathrm{new}}(576, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 2 + T )^{2}$$
$7$ $$48 + T^{2}$$
$11$ $$48 + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$( 10 + T )^{2}$$
$19$ $$432 + T^{2}$$
$23$ $$768 + T^{2}$$
$29$ $$( 26 + T )^{2}$$
$31$ $$48 + T^{2}$$
$37$ $$( 26 + T )^{2}$$
$41$ $$( 58 + T )^{2}$$
$43$ $$2352 + T^{2}$$
$47$ $$4800 + T^{2}$$
$53$ $$( 74 + T )^{2}$$
$59$ $$8112 + T^{2}$$
$61$ $$( 26 + T )^{2}$$
$67$ $$48 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 46 + T )^{2}$$
$79$ $$13872 + T^{2}$$
$83$ $$2352 + T^{2}$$
$89$ $$( 82 + T )^{2}$$
$97$ $$( -2 + T )^{2}$$