# Properties

 Label 2304.3.e.k 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: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 1152) 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.

 $$f(q)$$ $$=$$ $$q + 3 \beta_{1} q^{5} -\beta_{2} q^{7} +O(q^{10})$$ $$q + 3 \beta_{1} q^{5} -\beta_{2} q^{7} -\beta_{3} q^{11} + 10 q^{13} + 17 \beta_{1} q^{17} + 2 \beta_{2} q^{19} -\beta_{3} q^{23} + 7 q^{25} + 11 \beta_{1} q^{29} -\beta_{2} q^{31} -3 \beta_{3} q^{35} -64 q^{37} -9 \beta_{1} q^{41} + 4 \beta_{2} q^{43} -\beta_{3} q^{47} + 111 q^{49} -13 \beta_{1} q^{53} + 6 \beta_{2} q^{55} + 30 \beta_{1} q^{65} -6 \beta_{2} q^{67} + 7 \beta_{3} q^{71} -96 q^{73} + 160 \beta_{1} q^{77} + 5 \beta_{2} q^{79} + 7 \beta_{3} q^{83} -102 q^{85} -39 \beta_{1} q^{89} -10 \beta_{2} q^{91} + 6 \beta_{3} q^{95} + 64 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 40q^{13} + 28q^{25} - 256q^{37} + 444q^{49} - 384q^{73} - 408q^{85} + 256q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{3} + 28 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$8 \nu^{2} - 16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 4 \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 16$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{2} + 28 \beta_{1}$$$$)/8$$

## 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
 1.58114 − 0.707107i −1.58114 − 0.707107i 1.58114 + 0.707107i −1.58114 + 0.707107i
0 0 0 4.24264i 0 −12.6491 0 0 0
1025.2 0 0 0 4.24264i 0 12.6491 0 0 0
1025.3 0 0 0 4.24264i 0 −12.6491 0 0 0
1025.4 0 0 0 4.24264i 0 12.6491 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
4.b odd 2 1 inner
12.b 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.k 4
3.b odd 2 1 inner 2304.3.e.k 4
4.b odd 2 1 inner 2304.3.e.k 4
8.b even 2 1 2304.3.e.f 4
8.d odd 2 1 2304.3.e.f 4
12.b even 2 1 inner 2304.3.e.k 4
16.e even 4 2 1152.3.h.e 8
16.f odd 4 2 1152.3.h.e 8
24.f even 2 1 2304.3.e.f 4
24.h odd 2 1 2304.3.e.f 4
48.i odd 4 2 1152.3.h.e 8
48.k even 4 2 1152.3.h.e 8

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

## Hecke kernels

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

 $$T_{5}^{2} + 18$$ $$T_{7}^{2} - 160$$ $$T_{13} - 10$$ $$T_{19}^{2} - 640$$ $$T_{31}^{2} - 160$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 18 + T^{2} )^{2}$$
$7$ $$( -160 + T^{2} )^{2}$$
$11$ $$( 320 + T^{2} )^{2}$$
$13$ $$( -10 + T )^{4}$$
$17$ $$( 578 + T^{2} )^{2}$$
$19$ $$( -640 + T^{2} )^{2}$$
$23$ $$( 320 + T^{2} )^{2}$$
$29$ $$( 242 + T^{2} )^{2}$$
$31$ $$( -160 + T^{2} )^{2}$$
$37$ $$( 64 + T )^{4}$$
$41$ $$( 162 + T^{2} )^{2}$$
$43$ $$( -2560 + T^{2} )^{2}$$
$47$ $$( 320 + T^{2} )^{2}$$
$53$ $$( 338 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$( -5760 + T^{2} )^{2}$$
$71$ $$( 15680 + T^{2} )^{2}$$
$73$ $$( 96 + T )^{4}$$
$79$ $$( -4000 + T^{2} )^{2}$$
$83$ $$( 15680 + T^{2} )^{2}$$
$89$ $$( 3042 + T^{2} )^{2}$$
$97$ $$( -64 + T )^{4}$$