# Properties

 Label 2304.3.g.k Level $2304$ Weight $3$ Character orbit 2304.g Analytic conductor $62.779$ Analytic rank $0$ Dimension $2$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2304.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 128) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 12\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q +O(q^{10})$$ $$q + \beta q^{11} + 2 q^{17} + \beta q^{19} -25 q^{25} -46 q^{41} -5 \beta q^{43} + 49 q^{49} + 5 \beta q^{59} -7 \beta q^{67} -142 q^{73} + 3 \beta q^{83} -146 q^{89} + 94 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 4q^{17} - 50q^{25} - 92q^{41} + 98q^{49} - 284q^{73} - 292q^{89} + 188q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1279.1
 − 1.41421i 1.41421i
0 0 0 0 0 0 0 0 0
1279.2 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 2304.3.g.k 2
3.b odd 2 1 256.3.c.d 2
4.b odd 2 1 inner 2304.3.g.k 2
8.b even 2 1 inner 2304.3.g.k 2
8.d odd 2 1 CM 2304.3.g.k 2
12.b even 2 1 256.3.c.d 2
16.e even 4 2 1152.3.b.c 2
16.f odd 4 2 1152.3.b.c 2
24.f even 2 1 256.3.c.d 2
24.h odd 2 1 256.3.c.d 2
48.i odd 4 2 128.3.d.b 2
48.k even 4 2 128.3.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.3.d.b 2 48.i odd 4 2
128.3.d.b 2 48.k even 4 2
256.3.c.d 2 3.b odd 2 1
256.3.c.d 2 12.b even 2 1
256.3.c.d 2 24.f even 2 1
256.3.c.d 2 24.h odd 2 1
1152.3.b.c 2 16.e even 4 2
1152.3.b.c 2 16.f odd 4 2
2304.3.g.k 2 1.a even 1 1 trivial
2304.3.g.k 2 4.b odd 2 1 inner
2304.3.g.k 2 8.b even 2 1 inner
2304.3.g.k 2 8.d odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5}$$ $$T_{7}$$ $$T_{11}^{2} + 288$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$288 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$288 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$( 46 + T )^{2}$$
$43$ $$7200 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$7200 + T^{2}$$
$61$ $$T^{2}$$
$67$ $$14112 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 142 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$2592 + T^{2}$$
$89$ $$( 146 + T )^{2}$$
$97$ $$( -94 + T )^{2}$$