# Properties

 Label 2304.3.g.n 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: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 384) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 q^{5} + ( 4 - 8 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 4 q^{5} + ( 4 - 8 \zeta_{6} ) q^{7} + ( 4 - 8 \zeta_{6} ) q^{11} + 18 q^{17} + ( -12 + 24 \zeta_{6} ) q^{19} + ( -24 + 48 \zeta_{6} ) q^{23} -9 q^{25} -4 q^{29} + ( -28 + 56 \zeta_{6} ) q^{31} + ( 16 - 32 \zeta_{6} ) q^{35} + 72 q^{37} + 18 q^{41} + ( -36 + 72 \zeta_{6} ) q^{43} + ( 24 - 48 \zeta_{6} ) q^{47} + q^{49} + 44 q^{53} + ( 16 - 32 \zeta_{6} ) q^{55} + ( 36 - 72 \zeta_{6} ) q^{59} -72 q^{61} + ( -12 + 24 \zeta_{6} ) q^{67} + ( 24 - 48 \zeta_{6} ) q^{71} + 82 q^{73} -48 q^{77} + ( 36 - 72 \zeta_{6} ) q^{79} + ( 76 - 152 \zeta_{6} ) q^{83} + 72 q^{85} + 126 q^{89} + ( -48 + 96 \zeta_{6} ) q^{95} + 110 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{5} + O(q^{10})$$ $$2q + 8q^{5} + 36q^{17} - 18q^{25} - 8q^{29} + 144q^{37} + 36q^{41} + 2q^{49} + 88q^{53} - 144q^{61} + 164q^{73} - 96q^{77} + 144q^{85} + 252q^{89} + 220q^{97} + O(q^{100})$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 4.00000 0 6.92820i 0 0 0
1279.2 0 0 0 4.00000 0 6.92820i 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.n 2
3.b odd 2 1 768.3.g.a 2
4.b odd 2 1 inner 2304.3.g.n 2
8.b even 2 1 2304.3.g.g 2
8.d odd 2 1 2304.3.g.g 2
12.b even 2 1 768.3.g.a 2
16.e even 4 2 1152.3.b.h 4
16.f odd 4 2 1152.3.b.h 4
24.f even 2 1 768.3.g.b 2
24.h odd 2 1 768.3.g.b 2
48.i odd 4 2 384.3.b.a 4
48.k even 4 2 384.3.b.a 4

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

## Hecke kernels

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

 $$T_{5} - 4$$ $$T_{7}^{2} + 48$$ $$T_{11}^{2} + 48$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -4 + T )^{2}$$
$7$ $$48 + T^{2}$$
$11$ $$48 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( -18 + T )^{2}$$
$19$ $$432 + T^{2}$$
$23$ $$1728 + T^{2}$$
$29$ $$( 4 + T )^{2}$$
$31$ $$2352 + T^{2}$$
$37$ $$( -72 + T )^{2}$$
$41$ $$( -18 + T )^{2}$$
$43$ $$3888 + T^{2}$$
$47$ $$1728 + T^{2}$$
$53$ $$( -44 + T )^{2}$$
$59$ $$3888 + T^{2}$$
$61$ $$( 72 + T )^{2}$$
$67$ $$432 + T^{2}$$
$71$ $$1728 + T^{2}$$
$73$ $$( -82 + T )^{2}$$
$79$ $$3888 + T^{2}$$
$83$ $$17328 + T^{2}$$
$89$ $$( -126 + T )^{2}$$
$97$ $$( -110 + T )^{2}$$