# Properties

 Label 2304.3.g.i Level $2304$ Weight $3$ Character orbit 2304.g Analytic conductor $62.779$ Analytic rank $0$ Dimension $2$ CM discriminant -24 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: $$\Q(\sqrt{-6})$$ Defining polynomial: $$x^{2} + 6$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1152) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 4\sqrt{-6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{5} -\beta q^{7} +O(q^{10})$$ $$q -2 q^{5} -\beta q^{7} + 2 \beta q^{11} -21 q^{25} + 50 q^{29} -5 \beta q^{31} + 2 \beta q^{35} -47 q^{49} -94 q^{53} -4 \beta q^{55} -12 \beta q^{59} -50 q^{73} + 192 q^{77} + 15 \beta q^{79} -10 \beta q^{83} -190 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} + O(q^{10})$$ $$2q - 4q^{5} - 42q^{25} + 100q^{29} - 94q^{49} - 188q^{53} - 100q^{73} + 384q^{77} - 380q^{97} + O(q^{100})$$

## 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
 2.44949i − 2.44949i
0 0 0 −2.00000 0 9.79796i 0 0 0
1279.2 0 0 0 −2.00000 0 9.79796i 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 $$\Q(\sqrt{-6})$$
4.b odd 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.i 2
3.b odd 2 1 2304.3.g.l 2
4.b odd 2 1 inner 2304.3.g.i 2
8.b even 2 1 2304.3.g.l 2
8.d odd 2 1 2304.3.g.l 2
12.b even 2 1 2304.3.g.l 2
16.e even 4 2 1152.3.b.f 4
16.f odd 4 2 1152.3.b.f 4
24.f even 2 1 inner 2304.3.g.i 2
24.h odd 2 1 CM 2304.3.g.i 2
48.i odd 4 2 1152.3.b.f 4
48.k even 4 2 1152.3.b.f 4

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 2 + T )^{2}$$
$7$ $$96 + T^{2}$$
$11$ $$384 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -50 + T )^{2}$$
$31$ $$2400 + T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$( 94 + T )^{2}$$
$59$ $$13824 + T^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 50 + T )^{2}$$
$79$ $$21600 + T^{2}$$
$83$ $$9600 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$( 190 + T )^{2}$$