# Properties

 Label 2304.3.g.h Level $2304$ Weight $3$ Character orbit 2304.g Analytic conductor $62.779$ Analytic rank $0$ Dimension $2$ CM no 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: 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: $$2$$ Twist minimal: no (minimal twist has level 128) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -4 q^{5} -4 \beta q^{7} +O(q^{10})$$ $$q -4 q^{5} -4 \beta q^{7} + 5 \beta q^{11} + 20 q^{13} + 10 q^{17} + 5 \beta q^{19} -4 \beta q^{23} -9 q^{25} -20 q^{29} + 16 \beta q^{35} + 20 q^{37} -30 q^{41} -\beta q^{43} + 24 \beta q^{47} -79 q^{49} + 60 q^{53} -20 \beta q^{55} -15 \beta q^{59} -28 q^{61} -80 q^{65} + 29 \beta q^{67} + 20 \beta q^{71} + 10 q^{73} + 160 q^{77} + 40 \beta q^{79} -9 \beta q^{83} -40 q^{85} + 22 q^{89} -80 \beta q^{91} -20 \beta q^{95} + 150 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{5} + O(q^{10})$$ $$2q - 8q^{5} + 40q^{13} + 20q^{17} - 18q^{25} - 40q^{29} + 40q^{37} - 60q^{41} - 158q^{49} + 120q^{53} - 56q^{61} - 160q^{65} + 20q^{73} + 320q^{77} - 80q^{85} + 44q^{89} + 300q^{97} + O(q^{100})$$

## 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 −4.00000 0 11.3137i 0 0 0
1279.2 0 0 0 −4.00000 0 11.3137i 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 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{5} + 4$$ $$T_{7}^{2} + 128$$ $$T_{11}^{2} + 200$$ $$T_{13} - 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 4 + T )^{2}$$
$7$ $$128 + T^{2}$$
$11$ $$200 + T^{2}$$
$13$ $$( -20 + T )^{2}$$
$17$ $$( -10 + T )^{2}$$
$19$ $$200 + T^{2}$$
$23$ $$128 + T^{2}$$
$29$ $$( 20 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$( -20 + T )^{2}$$
$41$ $$( 30 + T )^{2}$$
$43$ $$8 + T^{2}$$
$47$ $$4608 + T^{2}$$
$53$ $$( -60 + T )^{2}$$
$59$ $$1800 + T^{2}$$
$61$ $$( 28 + T )^{2}$$
$67$ $$6728 + T^{2}$$
$71$ $$3200 + T^{2}$$
$73$ $$( -10 + T )^{2}$$
$79$ $$12800 + T^{2}$$
$83$ $$648 + T^{2}$$
$89$ $$( -22 + T )^{2}$$
$97$ $$( -150 + T )^{2}$$