# Properties

 Label 576.2.d.a Level $576$ Weight $2$ Character orbit 576.d Analytic conductor $4.599$ Analytic rank $0$ Dimension $2$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 576.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.59938315643$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 64) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q +O(q^{10})$$ $$q + 6 i q^{11} + 6 q^{17} + 2 i q^{19} + 5 q^{25} + 6 q^{41} + 10 i q^{43} -7 q^{49} + 6 i q^{59} -14 i q^{67} + 2 q^{73} -18 i q^{83} -18 q^{89} + 10 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 12q^{17} + 10q^{25} + 12q^{41} - 14q^{49} + 4q^{73} - 36q^{89} + 20q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 − 1.00000i 1.00000i
0 0 0 0 0 0 0 0 0
289.2 0 0 0 0 0 0 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.d.a 2
3.b odd 2 1 64.2.b.a 2
4.b odd 2 1 inner 576.2.d.a 2
8.b even 2 1 inner 576.2.d.a 2
8.d odd 2 1 CM 576.2.d.a 2
12.b even 2 1 64.2.b.a 2
15.d odd 2 1 1600.2.d.a 2
15.e even 4 1 1600.2.f.a 2
15.e even 4 1 1600.2.f.b 2
16.e even 4 1 2304.2.a.h 1
16.e even 4 1 2304.2.a.i 1
16.f odd 4 1 2304.2.a.h 1
16.f odd 4 1 2304.2.a.i 1
21.c even 2 1 3136.2.b.b 2
24.f even 2 1 64.2.b.a 2
24.h odd 2 1 64.2.b.a 2
48.i odd 4 1 256.2.a.a 1
48.i odd 4 1 256.2.a.d 1
48.k even 4 1 256.2.a.a 1
48.k even 4 1 256.2.a.d 1
60.h even 2 1 1600.2.d.a 2
60.l odd 4 1 1600.2.f.a 2
60.l odd 4 1 1600.2.f.b 2
84.h odd 2 1 3136.2.b.b 2
96.o even 8 4 1024.2.e.l 4
96.p odd 8 4 1024.2.e.l 4
120.i odd 2 1 1600.2.d.a 2
120.m even 2 1 1600.2.d.a 2
120.q odd 4 1 1600.2.f.a 2
120.q odd 4 1 1600.2.f.b 2
120.w even 4 1 1600.2.f.a 2
120.w even 4 1 1600.2.f.b 2
168.e odd 2 1 3136.2.b.b 2
168.i even 2 1 3136.2.b.b 2
240.t even 4 1 6400.2.a.a 1
240.t even 4 1 6400.2.a.x 1
240.bm odd 4 1 6400.2.a.a 1
240.bm odd 4 1 6400.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 3.b odd 2 1
64.2.b.a 2 12.b even 2 1
64.2.b.a 2 24.f even 2 1
64.2.b.a 2 24.h odd 2 1
256.2.a.a 1 48.i odd 4 1
256.2.a.a 1 48.k even 4 1
256.2.a.d 1 48.i odd 4 1
256.2.a.d 1 48.k even 4 1
576.2.d.a 2 1.a even 1 1 trivial
576.2.d.a 2 4.b odd 2 1 inner
576.2.d.a 2 8.b even 2 1 inner
576.2.d.a 2 8.d odd 2 1 CM
1024.2.e.l 4 96.o even 8 4
1024.2.e.l 4 96.p odd 8 4
1600.2.d.a 2 15.d odd 2 1
1600.2.d.a 2 60.h even 2 1
1600.2.d.a 2 120.i odd 2 1
1600.2.d.a 2 120.m even 2 1
1600.2.f.a 2 15.e even 4 1
1600.2.f.a 2 60.l odd 4 1
1600.2.f.a 2 120.q odd 4 1
1600.2.f.a 2 120.w even 4 1
1600.2.f.b 2 15.e even 4 1
1600.2.f.b 2 60.l odd 4 1
1600.2.f.b 2 120.q odd 4 1
1600.2.f.b 2 120.w even 4 1
2304.2.a.h 1 16.e even 4 1
2304.2.a.h 1 16.f odd 4 1
2304.2.a.i 1 16.e even 4 1
2304.2.a.i 1 16.f odd 4 1
3136.2.b.b 2 21.c even 2 1
3136.2.b.b 2 84.h odd 2 1
3136.2.b.b 2 168.e odd 2 1
3136.2.b.b 2 168.i even 2 1
6400.2.a.a 1 240.t even 4 1
6400.2.a.a 1 240.bm odd 4 1
6400.2.a.x 1 240.t even 4 1
6400.2.a.x 1 240.bm odd 4 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$36 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$4 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$100 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$36 + T^{2}$$
$61$ $$T^{2}$$
$67$ $$196 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$324 + T^{2}$$
$89$ $$( 18 + T )^{2}$$
$97$ $$( -10 + T )^{2}$$