# Properties

 Label 2304.3.g.t Level $2304$ Weight $3$ Character orbit 2304.g Analytic conductor $62.779$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $8$

# 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 576) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} -5 \beta_{1} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} -5 \beta_{1} q^{7} + \beta_{3} q^{11} + 71 q^{25} -3 \beta_{2} q^{29} + 19 \beta_{1} q^{31} + 5 \beta_{3} q^{35} -51 q^{49} + 5 \beta_{2} q^{53} -96 \beta_{1} q^{55} + 6 \beta_{3} q^{59} + 50 q^{73} + 20 \beta_{2} q^{77} -29 \beta_{1} q^{79} -5 \beta_{3} q^{83} -190 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 284q^{25} - 204q^{49} + 200q^{73} - 760q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{2}$$$$/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{3} + 12 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$8 \nu^{3} + 24 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{1}$$$$/2$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{3} - 6 \beta_{2}$$$$)/16$$

## 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
 1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i −1.22474 + 1.22474i
0 0 0 −9.79796 0 10.0000i 0 0 0
1279.2 0 0 0 −9.79796 0 10.0000i 0 0 0
1279.3 0 0 0 9.79796 0 10.0000i 0 0 0
1279.4 0 0 0 9.79796 0 10.0000i 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f 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.t 4
3.b odd 2 1 inner 2304.3.g.t 4
4.b odd 2 1 inner 2304.3.g.t 4
8.b even 2 1 inner 2304.3.g.t 4
8.d odd 2 1 inner 2304.3.g.t 4
12.b even 2 1 inner 2304.3.g.t 4
16.e even 4 2 576.3.b.b 4
16.f odd 4 2 576.3.b.b 4
24.f even 2 1 inner 2304.3.g.t 4
24.h odd 2 1 CM 2304.3.g.t 4
48.i odd 4 2 576.3.b.b 4
48.k even 4 2 576.3.b.b 4

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

 $$T_{5}^{2} - 96$$ $$T_{7}^{2} + 100$$ $$T_{11}^{2} + 384$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -96 + T^{2} )^{2}$$
$7$ $$( 100 + T^{2} )^{2}$$
$11$ $$( 384 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( -864 + T^{2} )^{2}$$
$31$ $$( 1444 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( -2400 + T^{2} )^{2}$$
$59$ $$( 13824 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -50 + T )^{4}$$
$79$ $$( 3364 + T^{2} )^{2}$$
$83$ $$( 9600 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$( 190 + T )^{4}$$