# Properties

 Label 2304.3.h.g Level $2304$ Weight $3$ Character orbit 2304.h Analytic conductor $62.779$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2304.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 288) 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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} + 8 q^{7} +O(q^{10})$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} + 8 q^{7} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{11} + 8 \zeta_{8}^{2} q^{13} + ( 9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{17} + 32 \zeta_{8}^{2} q^{19} + ( -24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{23} -23 q^{25} + ( -31 \zeta_{8} + 31 \zeta_{8}^{3} ) q^{29} -40 q^{31} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{35} -26 \zeta_{8}^{2} q^{37} + ( -47 \zeta_{8} - 47 \zeta_{8}^{3} ) q^{41} -16 \zeta_{8}^{2} q^{43} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{47} + 15 q^{49} + ( 23 \zeta_{8} - 23 \zeta_{8}^{3} ) q^{53} -16 q^{55} + ( -16 \zeta_{8} + 16 \zeta_{8}^{3} ) q^{59} + 54 \zeta_{8}^{2} q^{61} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{65} -80 \zeta_{8}^{2} q^{67} + ( 56 \zeta_{8} + 56 \zeta_{8}^{3} ) q^{71} -96 q^{73} + ( -64 \zeta_{8} + 64 \zeta_{8}^{3} ) q^{77} + 104 q^{79} + ( -72 \zeta_{8} + 72 \zeta_{8}^{3} ) q^{83} + 18 \zeta_{8}^{2} q^{85} + ( -55 \zeta_{8} - 55 \zeta_{8}^{3} ) q^{89} + 64 \zeta_{8}^{2} q^{91} + ( 32 \zeta_{8} + 32 \zeta_{8}^{3} ) q^{95} -80 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 32q^{7} + O(q^{10})$$ $$4q + 32q^{7} - 92q^{25} - 160q^{31} + 60q^{49} - 64q^{55} - 384q^{73} + 416q^{79} - 320q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2177.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
0 0 0 −1.41421 0 8.00000 0 0 0
2177.2 0 0 0 −1.41421 0 8.00000 0 0 0
2177.3 0 0 0 1.41421 0 8.00000 0 0 0
2177.4 0 0 0 1.41421 0 8.00000 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
3.b odd 2 1 inner
8.b even 2 1 inner
24.h 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.h.g 4
3.b odd 2 1 inner 2304.3.h.g 4
4.b odd 2 1 2304.3.h.b 4
8.b even 2 1 inner 2304.3.h.g 4
8.d odd 2 1 2304.3.h.b 4
12.b even 2 1 2304.3.h.b 4
16.e even 4 1 288.3.e.a 2
16.e even 4 1 576.3.e.b 2
16.f odd 4 1 288.3.e.d yes 2
16.f odd 4 1 576.3.e.g 2
24.f even 2 1 2304.3.h.b 4
24.h odd 2 1 inner 2304.3.h.g 4
48.i odd 4 1 288.3.e.a 2
48.i odd 4 1 576.3.e.b 2
48.k even 4 1 288.3.e.d yes 2
48.k even 4 1 576.3.e.g 2

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

 $$T_{5}^{2} - 2$$ $$T_{7} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -2 + T^{2} )^{2}$$
$7$ $$( -8 + T )^{4}$$
$11$ $$( -128 + T^{2} )^{2}$$
$13$ $$( 64 + T^{2} )^{2}$$
$17$ $$( 162 + T^{2} )^{2}$$
$19$ $$( 1024 + T^{2} )^{2}$$
$23$ $$( 1152 + T^{2} )^{2}$$
$29$ $$( -1922 + T^{2} )^{2}$$
$31$ $$( 40 + T )^{4}$$
$37$ $$( 676 + T^{2} )^{2}$$
$41$ $$( 4418 + T^{2} )^{2}$$
$43$ $$( 256 + T^{2} )^{2}$$
$47$ $$( 128 + T^{2} )^{2}$$
$53$ $$( -1058 + T^{2} )^{2}$$
$59$ $$( -512 + T^{2} )^{2}$$
$61$ $$( 2916 + T^{2} )^{2}$$
$67$ $$( 6400 + T^{2} )^{2}$$
$71$ $$( 6272 + T^{2} )^{2}$$
$73$ $$( 96 + T )^{4}$$
$79$ $$( -104 + T )^{4}$$
$83$ $$( -10368 + T^{2} )^{2}$$
$89$ $$( 6050 + T^{2} )^{2}$$
$97$ $$( 80 + T )^{4}$$
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