# Properties

 Label 2304.3.g.q Level $2304$ Weight $3$ Character orbit 2304.g Analytic conductor $62.779$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 192) 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} -10 \zeta_{12}^{3} q^{7} +O(q^{10})$$ $$q + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} -10 \zeta_{12}^{3} q^{7} + ( -8 + 16 \zeta_{12}^{2} ) q^{11} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{13} -30 q^{17} + ( -4 + 8 \zeta_{12}^{2} ) q^{19} + 12 \zeta_{12}^{3} q^{23} -13 q^{25} + ( 60 \zeta_{12} - 30 \zeta_{12}^{3} ) q^{29} + 14 \zeta_{12}^{3} q^{31} + ( 20 - 40 \zeta_{12}^{2} ) q^{35} + ( -64 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{37} -6 q^{41} + ( -36 + 72 \zeta_{12}^{2} ) q^{43} + 84 \zeta_{12}^{3} q^{47} -51 q^{49} + ( -20 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{53} + 48 \zeta_{12}^{3} q^{55} + ( -36 + 72 \zeta_{12}^{2} ) q^{59} + ( -112 \zeta_{12} + 56 \zeta_{12}^{3} ) q^{61} + 24 q^{65} + ( 28 - 56 \zeta_{12}^{2} ) q^{67} -60 \zeta_{12}^{3} q^{71} + 86 q^{73} + ( 160 \zeta_{12} - 80 \zeta_{12}^{3} ) q^{77} + 38 \zeta_{12}^{3} q^{79} + ( -8 + 16 \zeta_{12}^{2} ) q^{83} + ( -120 \zeta_{12} + 60 \zeta_{12}^{3} ) q^{85} + 78 q^{89} + ( 40 - 80 \zeta_{12}^{2} ) q^{91} + 24 \zeta_{12}^{3} q^{95} + 62 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 120q^{17} - 52q^{25} - 24q^{41} - 204q^{49} + 96q^{65} + 344q^{73} + 312q^{89} + 248q^{97} + O(q^{100})$$

## 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
 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i
0 0 0 −3.46410 0 10.0000i 0 0 0
1279.2 0 0 0 −3.46410 0 10.0000i 0 0 0
1279.3 0 0 0 3.46410 0 10.0000i 0 0 0
1279.4 0 0 0 3.46410 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d 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.q 4
3.b odd 2 1 768.3.g.f 4
4.b odd 2 1 inner 2304.3.g.q 4
8.b even 2 1 inner 2304.3.g.q 4
8.d odd 2 1 inner 2304.3.g.q 4
12.b even 2 1 768.3.g.f 4
16.e even 4 2 576.3.b.a 4
16.f odd 4 2 576.3.b.a 4
24.f even 2 1 768.3.g.f 4
24.h odd 2 1 768.3.g.f 4
48.i odd 4 2 192.3.b.b 4
48.k even 4 2 192.3.b.b 4

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

 $$T_{5}^{2} - 12$$ $$T_{7}^{2} + 100$$ $$T_{11}^{2} + 192$$ $$T_{13}^{2} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -12 + T^{2} )^{2}$$
$7$ $$( 100 + T^{2} )^{2}$$
$11$ $$( 192 + T^{2} )^{2}$$
$13$ $$( -48 + T^{2} )^{2}$$
$17$ $$( 30 + T )^{4}$$
$19$ $$( 48 + T^{2} )^{2}$$
$23$ $$( 144 + T^{2} )^{2}$$
$29$ $$( -2700 + T^{2} )^{2}$$
$31$ $$( 196 + T^{2} )^{2}$$
$37$ $$( -3072 + T^{2} )^{2}$$
$41$ $$( 6 + T )^{4}$$
$43$ $$( 3888 + T^{2} )^{2}$$
$47$ $$( 7056 + T^{2} )^{2}$$
$53$ $$( -300 + T^{2} )^{2}$$
$59$ $$( 3888 + T^{2} )^{2}$$
$61$ $$( -9408 + T^{2} )^{2}$$
$67$ $$( 2352 + T^{2} )^{2}$$
$71$ $$( 3600 + T^{2} )^{2}$$
$73$ $$( -86 + T )^{4}$$
$79$ $$( 1444 + T^{2} )^{2}$$
$83$ $$( 192 + T^{2} )^{2}$$
$89$ $$( -78 + T )^{4}$$
$97$ $$( -62 + T )^{4}$$