# Properties

 Label 2304.3.e.g Level $2304$ Weight $3$ Character orbit 2304.e 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.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ Defining polynomial: $$x^{4} + 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 576) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} -\beta_{3} q^{7} +O(q^{10})$$ $$q + \beta_{2} q^{5} -\beta_{3} q^{7} + 2 \beta_{1} q^{11} + \beta_{3} q^{13} + 5 \beta_{1} q^{17} -20 q^{19} + 2 \beta_{2} q^{23} -29 q^{25} + 5 \beta_{2} q^{29} -5 \beta_{3} q^{31} -18 \beta_{1} q^{35} + 4 \beta_{3} q^{37} -17 \beta_{1} q^{41} + 40 q^{43} + 10 \beta_{2} q^{47} + 59 q^{49} + 5 \beta_{2} q^{53} -6 \beta_{3} q^{55} + 8 \beta_{1} q^{59} + 18 \beta_{1} q^{65} -100 q^{67} + 10 \beta_{2} q^{71} -20 q^{73} -12 \beta_{2} q^{77} + 5 \beta_{3} q^{79} -30 \beta_{1} q^{83} -15 \beta_{3} q^{85} -3 \beta_{1} q^{89} -108 q^{91} -20 \beta_{2} q^{95} + 40 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 80q^{19} - 116q^{25} + 160q^{43} + 236q^{49} - 400q^{67} - 80q^{73} - 432q^{91} + 160q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$3 \nu^{3} + 9 \nu$$ $$\beta_{2}$$ $$=$$ $$3 \nu^{3} + 15 \nu$$ $$\beta_{3}$$ $$=$$ $$6 \nu^{2} + 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 12$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{2} + 5 \beta_{1}$$$$)/6$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 − 0.517638i − 1.93185i 0.517638i 1.93185i
0 0 0 7.34847i 0 −10.3923 0 0 0
1025.2 0 0 0 7.34847i 0 10.3923 0 0 0
1025.3 0 0 0 7.34847i 0 −10.3923 0 0 0
1025.4 0 0 0 7.34847i 0 10.3923 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.d 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.e.g 4
3.b odd 2 1 inner 2304.3.e.g 4
4.b odd 2 1 2304.3.e.j 4
8.b even 2 1 2304.3.e.j 4
8.d odd 2 1 inner 2304.3.e.g 4
12.b even 2 1 2304.3.e.j 4
16.e even 4 2 576.3.h.b 8
16.f odd 4 2 576.3.h.b 8
24.f even 2 1 inner 2304.3.e.g 4
24.h odd 2 1 2304.3.e.j 4
48.i odd 4 2 576.3.h.b 8
48.k even 4 2 576.3.h.b 8

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

## Hecke kernels

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

 $$T_{5}^{2} + 54$$ $$T_{7}^{2} - 108$$ $$T_{13}^{2} - 108$$ $$T_{19} + 20$$ $$T_{31}^{2} - 2700$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 54 + T^{2} )^{2}$$
$7$ $$( -108 + T^{2} )^{2}$$
$11$ $$( 72 + T^{2} )^{2}$$
$13$ $$( -108 + T^{2} )^{2}$$
$17$ $$( 450 + T^{2} )^{2}$$
$19$ $$( 20 + T )^{4}$$
$23$ $$( 216 + T^{2} )^{2}$$
$29$ $$( 1350 + T^{2} )^{2}$$
$31$ $$( -2700 + T^{2} )^{2}$$
$37$ $$( -1728 + T^{2} )^{2}$$
$41$ $$( 5202 + T^{2} )^{2}$$
$43$ $$( -40 + T )^{4}$$
$47$ $$( 5400 + T^{2} )^{2}$$
$53$ $$( 1350 + T^{2} )^{2}$$
$59$ $$( 1152 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$( 100 + T )^{4}$$
$71$ $$( 5400 + T^{2} )^{2}$$
$73$ $$( 20 + T )^{4}$$
$79$ $$( -2700 + T^{2} )^{2}$$
$83$ $$( 16200 + T^{2} )^{2}$$
$89$ $$( 162 + T^{2} )^{2}$$
$97$ $$( -40 + T )^{4}$$