# Properties

 Label 288.3.e.c Level $288$ Weight $3$ Character orbit 288.e Analytic conductor $7.847$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# 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: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} +O(q^{10})$$ $$q + \beta q^{5} + 24 q^{13} + 23 \beta q^{17} + 23 q^{25} -\beta q^{29} + 70 q^{37} + 49 \beta q^{41} -49 q^{49} -73 \beta q^{53} -22 q^{61} + 24 \beta q^{65} -96 q^{73} -46 q^{85} -119 \beta q^{89} -144 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 48q^{13} + 46q^{25} + 140q^{37} - 98q^{49} - 44q^{61} - 192q^{73} - 92q^{85} - 288q^{97} + O(q^{100})$$

## 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 0 0 0 0
161.2 0 0 0 1.41421i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
12.b even 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.c 2
3.b odd 2 1 inner 288.3.e.c 2
4.b odd 2 1 CM 288.3.e.c 2
8.b even 2 1 576.3.e.d 2
8.d odd 2 1 576.3.e.d 2
12.b even 2 1 inner 288.3.e.c 2
16.e even 4 2 2304.3.h.d 4
16.f odd 4 2 2304.3.h.d 4
24.f even 2 1 576.3.e.d 2
24.h odd 2 1 576.3.e.d 2
48.i odd 4 2 2304.3.h.d 4
48.k even 4 2 2304.3.h.d 4

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$2 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -24 + T )^{2}$$
$17$ $$1058 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$2 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( -70 + T )^{2}$$
$41$ $$4802 + T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$10658 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 22 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 96 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$28322 + T^{2}$$
$97$ $$( 144 + T )^{2}$$