# Properties

 Label 2304.1.g.b Level $2304$ Weight $1$ Character orbit 2304.g Self dual yes Analytic conductor $1.150$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -4, -8, 8 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$1.14984578911$$ 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) Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\zeta_{8})$$ Artin image $D_4$ Artin field Galois closure of 4.2.18432.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + O(q^{10})$$ $$q + 2q^{17} - q^{25} + 2q^{41} + q^{49} + 2q^{73} - 2q^{89} - 2q^{97} + O(q^{100})$$

## 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 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
8.b even 2 1 RM by $$\Q(\sqrt{2})$$
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.1.g.b 1
3.b odd 2 1 256.1.c.a 1
4.b odd 2 1 CM 2304.1.g.b 1
8.b even 2 1 RM 2304.1.g.b 1
8.d odd 2 1 CM 2304.1.g.b 1
12.b even 2 1 256.1.c.a 1
16.e even 4 2 1152.1.b.a 1
16.f odd 4 2 1152.1.b.a 1
24.f even 2 1 256.1.c.a 1
24.h odd 2 1 256.1.c.a 1
48.i odd 4 2 128.1.d.a 1
48.k even 4 2 128.1.d.a 1
96.o even 8 4 1024.1.f.b 2
96.p odd 8 4 1024.1.f.b 2
240.t even 4 2 3200.1.g.a 1
240.z odd 4 2 3200.1.e.a 2
240.bb even 4 2 3200.1.e.a 2
240.bd odd 4 2 3200.1.e.a 2
240.bf even 4 2 3200.1.e.a 2
240.bm odd 4 2 3200.1.g.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.1.d.a 1 48.i odd 4 2
128.1.d.a 1 48.k even 4 2
256.1.c.a 1 3.b odd 2 1
256.1.c.a 1 12.b even 2 1
256.1.c.a 1 24.f even 2 1
256.1.c.a 1 24.h odd 2 1
1024.1.f.b 2 96.o even 8 4
1024.1.f.b 2 96.p odd 8 4
1152.1.b.a 1 16.e even 4 2
1152.1.b.a 1 16.f odd 4 2
2304.1.g.b 1 1.a even 1 1 trivial
2304.1.g.b 1 4.b odd 2 1 CM
2304.1.g.b 1 8.b even 2 1 RM
2304.1.g.b 1 8.d odd 2 1 CM
3200.1.e.a 2 240.z odd 4 2
3200.1.e.a 2 240.bb even 4 2
3200.1.e.a 2 240.bd odd 4 2
3200.1.e.a 2 240.bf even 4 2
3200.1.g.a 1 240.t even 4 2
3200.1.g.a 1 240.bm odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2304, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$-2 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$-2 + T$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$-2 + T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$2 + T$$
$97$ $$2 + T$$