# Properties

 Label 2304.3.g.f Level $2304$ Weight $3$ Character orbit 2304.g Self dual yes Analytic conductor $62.779$ Analytic rank $0$ Dimension $1$ CM discriminant -4 Inner twists $2$

# 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: yes Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 128) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 8q^{5} + O(q^{10})$$ $$q + 8q^{5} + 24q^{13} - 30q^{17} + 39q^{25} + 40q^{29} + 24q^{37} + 18q^{41} + 49q^{49} - 56q^{53} + 120q^{61} + 192q^{65} - 110q^{73} - 240q^{85} + 78q^{89} - 130q^{97} + O(q^{100})$$

## 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
 0
0 0 0 8.00000 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 $$\Q(\sqrt{-1})$$

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.3.d.a 2 48.i odd 4 2
128.3.d.a 2 48.k even 4 2
256.3.c.a 1 3.b odd 2 1
256.3.c.a 1 12.b even 2 1
256.3.c.b 1 24.f even 2 1
256.3.c.b 1 24.h odd 2 1
1152.3.b.a 2 16.e even 4 2
1152.3.b.a 2 16.f odd 4 2
2304.3.g.a 1 8.b even 2 1
2304.3.g.a 1 8.d odd 2 1
2304.3.g.f 1 1.a even 1 1 trivial
2304.3.g.f 1 4.b 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 $$S_{3}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5} - 8$$ $$T_{7}$$ $$T_{11}$$ $$T_{13} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-8 + T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$-24 + T$$
$17$ $$30 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$-40 + T$$
$31$ $$T$$
$37$ $$-24 + T$$
$41$ $$-18 + T$$
$43$ $$T$$
$47$ $$T$$
$53$ $$56 + T$$
$59$ $$T$$
$61$ $$-120 + T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$110 + T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$-78 + T$$
$97$ $$130 + T$$