# Properties

 Label 288.3.e.a Level $288$ Weight $3$ Character orbit 288.e Analytic conductor $7.847$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 288.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.84743161358$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta q^{5} -8 q^{7} +O(q^{10})$$ $$q + \beta q^{5} -8 q^{7} -8 \beta q^{11} -8 q^{13} -9 \beta q^{17} -32 q^{19} -24 \beta q^{23} + 23 q^{25} + 31 \beta q^{29} -40 q^{31} -8 \beta q^{35} -26 q^{37} -47 \beta q^{41} -16 q^{43} + 8 \beta q^{47} + 15 q^{49} + 23 \beta q^{53} + 16 q^{55} -16 \beta q^{59} -54 q^{61} -8 \beta q^{65} + 80 q^{67} + 56 \beta q^{71} + 96 q^{73} + 64 \beta q^{77} + 104 q^{79} + 72 \beta q^{83} + 18 q^{85} -55 \beta q^{89} + 64 q^{91} -32 \beta q^{95} -80 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 16q^{7} + O(q^{10})$$ $$2q - 16q^{7} - 16q^{13} - 64q^{19} + 46q^{25} - 80q^{31} - 52q^{37} - 32q^{43} + 30q^{49} + 32q^{55} - 108q^{61} + 160q^{67} + 192q^{73} + 208q^{79} + 36q^{85} + 128q^{91} - 160q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 − 1.41421i 1.41421i
0 0 0 1.41421i 0 −8.00000 0 0 0
161.2 0 0 0 1.41421i 0 −8.00000 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 288.3.e.a 2
3.b odd 2 1 inner 288.3.e.a 2
4.b odd 2 1 288.3.e.d yes 2
8.b even 2 1 576.3.e.b 2
8.d odd 2 1 576.3.e.g 2
12.b even 2 1 288.3.e.d yes 2
16.e even 4 2 2304.3.h.g 4
16.f odd 4 2 2304.3.h.b 4
24.f even 2 1 576.3.e.g 2
24.h odd 2 1 576.3.e.b 2
48.i odd 4 2 2304.3.h.g 4
48.k even 4 2 2304.3.h.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.e.a 2 1.a even 1 1 trivial
288.3.e.a 2 3.b odd 2 1 inner
288.3.e.d yes 2 4.b odd 2 1
288.3.e.d yes 2 12.b even 2 1
576.3.e.b 2 8.b even 2 1
576.3.e.b 2 24.h odd 2 1
576.3.e.g 2 8.d odd 2 1
576.3.e.g 2 24.f even 2 1
2304.3.h.b 4 16.f odd 4 2
2304.3.h.b 4 48.k even 4 2
2304.3.h.g 4 16.e even 4 2
2304.3.h.g 4 48.i odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(288, [\chi])$$:

 $$T_{5}^{2} + 2$$ $$T_{7} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$2 + T^{2}$$
$7$ $$( 8 + T )^{2}$$
$11$ $$128 + T^{2}$$
$13$ $$( 8 + T )^{2}$$
$17$ $$162 + T^{2}$$
$19$ $$( 32 + T )^{2}$$
$23$ $$1152 + T^{2}$$
$29$ $$1922 + T^{2}$$
$31$ $$( 40 + T )^{2}$$
$37$ $$( 26 + T )^{2}$$
$41$ $$4418 + T^{2}$$
$43$ $$( 16 + T )^{2}$$
$47$ $$128 + T^{2}$$
$53$ $$1058 + T^{2}$$
$59$ $$512 + T^{2}$$
$61$ $$( 54 + T )^{2}$$
$67$ $$( -80 + T )^{2}$$
$71$ $$6272 + T^{2}$$
$73$ $$( -96 + T )^{2}$$
$79$ $$( -104 + T )^{2}$$
$83$ $$10368 + T^{2}$$
$89$ $$6050 + T^{2}$$
$97$ $$( 80 + T )^{2}$$