# Properties

 Label 2304.3.g.r 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: $$2^{8}\cdot 3$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{5} -12 \zeta_{12}^{3} q^{7} +O(q^{10})$$ $$q + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{5} -12 \zeta_{12}^{3} q^{7} + ( 4 - 8 \zeta_{12}^{2} ) q^{11} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{13} -14 q^{17} + ( 20 - 40 \zeta_{12}^{2} ) q^{19} -24 \zeta_{12}^{3} q^{23} + 23 q^{25} + ( 40 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{29} -12 \zeta_{12}^{3} q^{31} + ( -48 + 96 \zeta_{12}^{2} ) q^{35} + ( 32 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{37} -14 q^{41} + ( -4 + 8 \zeta_{12}^{2} ) q^{43} -72 \zeta_{12}^{3} q^{47} -95 q^{49} + ( 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{53} + 48 \zeta_{12}^{3} q^{55} + ( -28 + 56 \zeta_{12}^{2} ) q^{59} + ( 64 \zeta_{12} - 32 \zeta_{12}^{3} ) q^{61} -96 q^{65} + ( 52 - 104 \zeta_{12}^{2} ) q^{67} + 24 \zeta_{12}^{3} q^{71} + 50 q^{73} + ( -96 \zeta_{12} + 48 \zeta_{12}^{3} ) q^{77} -12 \zeta_{12}^{3} q^{79} + ( 12 - 24 \zeta_{12}^{2} ) q^{83} + ( 112 \zeta_{12} - 56 \zeta_{12}^{3} ) q^{85} + 62 q^{89} + ( 96 - 192 \zeta_{12}^{2} ) q^{91} + 240 \zeta_{12}^{3} q^{95} -146 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 56q^{17} + 92q^{25} - 56q^{41} - 380q^{49} - 384q^{65} + 200q^{73} + 248q^{89} - 584q^{97} + O(q^{100})$$

## 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 −6.92820 0 12.0000i 0 0 0
1279.2 0 0 0 −6.92820 0 12.0000i 0 0 0
1279.3 0 0 0 6.92820 0 12.0000i 0 0 0
1279.4 0 0 0 6.92820 0 12.0000i 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.r 4
3.b odd 2 1 768.3.g.e 4
4.b odd 2 1 inner 2304.3.g.r 4
8.b even 2 1 inner 2304.3.g.r 4
8.d odd 2 1 inner 2304.3.g.r 4
12.b even 2 1 768.3.g.e 4
16.e even 4 2 1152.3.b.e 4
16.f odd 4 2 1152.3.b.e 4
24.f even 2 1 768.3.g.e 4
24.h odd 2 1 768.3.g.e 4
48.i odd 4 2 384.3.b.b 4
48.k even 4 2 384.3.b.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.b.b 4 48.i odd 4 2
384.3.b.b 4 48.k even 4 2
768.3.g.e 4 3.b odd 2 1
768.3.g.e 4 12.b even 2 1
768.3.g.e 4 24.f even 2 1
768.3.g.e 4 24.h odd 2 1
1152.3.b.e 4 16.e even 4 2
1152.3.b.e 4 16.f odd 4 2
2304.3.g.r 4 1.a even 1 1 trivial
2304.3.g.r 4 4.b odd 2 1 inner
2304.3.g.r 4 8.b even 2 1 inner
2304.3.g.r 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 $$S_{3}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5}^{2} - 48$$ $$T_{7}^{2} + 144$$ $$T_{11}^{2} + 48$$ $$T_{13}^{2} - 192$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -48 + T^{2} )^{2}$$
$7$ $$( 144 + T^{2} )^{2}$$
$11$ $$( 48 + T^{2} )^{2}$$
$13$ $$( -192 + T^{2} )^{2}$$
$17$ $$( 14 + T )^{4}$$
$19$ $$( 1200 + T^{2} )^{2}$$
$23$ $$( 576 + T^{2} )^{2}$$
$29$ $$( -1200 + T^{2} )^{2}$$
$31$ $$( 144 + T^{2} )^{2}$$
$37$ $$( -768 + T^{2} )^{2}$$
$41$ $$( 14 + T )^{4}$$
$43$ $$( 48 + T^{2} )^{2}$$
$47$ $$( 5184 + T^{2} )^{2}$$
$53$ $$( -3888 + T^{2} )^{2}$$
$59$ $$( 2352 + T^{2} )^{2}$$
$61$ $$( -3072 + T^{2} )^{2}$$
$67$ $$( 8112 + T^{2} )^{2}$$
$71$ $$( 576 + T^{2} )^{2}$$
$73$ $$( -50 + T )^{4}$$
$79$ $$( 144 + T^{2} )^{2}$$
$83$ $$( 432 + T^{2} )^{2}$$
$89$ $$( -62 + T )^{4}$$
$97$ $$( 146 + T )^{4}$$