# Properties

 Label 2304.3.e.e 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: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ 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 primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{5} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{7} +O(q^{10})$$ $$q + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{5} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{7} -4 \zeta_{8}^{2} q^{11} -18 q^{13} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{17} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{19} -36 \zeta_{8}^{2} q^{23} + 7 q^{25} + ( -9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{29} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{31} + 36 \zeta_{8}^{2} q^{35} -36 q^{37} + ( -21 \zeta_{8} - 21 \zeta_{8}^{3} ) q^{41} + ( -48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{43} -36 \zeta_{8}^{2} q^{47} + 23 q^{49} + ( -57 \zeta_{8} - 57 \zeta_{8}^{3} ) q^{53} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{55} -80 \zeta_{8}^{2} q^{59} + 36 q^{61} + ( -54 \zeta_{8} - 54 \zeta_{8}^{3} ) q^{65} + ( 84 \zeta_{8} - 84 \zeta_{8}^{3} ) q^{67} + 108 \zeta_{8}^{2} q^{71} + 56 q^{73} + ( -24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{77} + ( 18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{79} + 76 \zeta_{8}^{2} q^{83} + 18 q^{85} + ( -63 \zeta_{8} - 63 \zeta_{8}^{3} ) q^{89} + ( -108 \zeta_{8} + 108 \zeta_{8}^{3} ) q^{91} + 72 \zeta_{8}^{2} q^{95} + 104 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 72q^{13} + 28q^{25} - 144q^{37} + 92q^{49} + 144q^{61} + 224q^{73} + 72q^{85} + 416q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
0 0 0 4.24264i 0 −8.48528 0 0 0
1025.2 0 0 0 4.24264i 0 8.48528 0 0 0
1025.3 0 0 0 4.24264i 0 −8.48528 0 0 0
1025.4 0 0 0 4.24264i 0 8.48528 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.e 4
3.b odd 2 1 inner 2304.3.e.e 4
4.b odd 2 1 inner 2304.3.e.e 4
8.b even 2 1 2304.3.e.l 4
8.d odd 2 1 2304.3.e.l 4
12.b even 2 1 inner 2304.3.e.e 4
16.e even 4 1 1152.3.h.a 4
16.e even 4 1 1152.3.h.d yes 4
16.f odd 4 1 1152.3.h.a 4
16.f odd 4 1 1152.3.h.d yes 4
24.f even 2 1 2304.3.e.l 4
24.h odd 2 1 2304.3.e.l 4
48.i odd 4 1 1152.3.h.a 4
48.i odd 4 1 1152.3.h.d yes 4
48.k even 4 1 1152.3.h.a 4
48.k even 4 1 1152.3.h.d yes 4

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

 $$T_{5}^{2} + 18$$ $$T_{7}^{2} - 72$$ $$T_{13} + 18$$ $$T_{19}^{2} - 288$$ $$T_{31}^{2} - 72$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 18 + T^{2} )^{2}$$
$7$ $$( -72 + T^{2} )^{2}$$
$11$ $$( 16 + T^{2} )^{2}$$
$13$ $$( 18 + T )^{4}$$
$17$ $$( 18 + T^{2} )^{2}$$
$19$ $$( -288 + T^{2} )^{2}$$
$23$ $$( 1296 + T^{2} )^{2}$$
$29$ $$( 162 + T^{2} )^{2}$$
$31$ $$( -72 + T^{2} )^{2}$$
$37$ $$( 36 + T )^{4}$$
$41$ $$( 882 + T^{2} )^{2}$$
$43$ $$( -4608 + T^{2} )^{2}$$
$47$ $$( 1296 + T^{2} )^{2}$$
$53$ $$( 6498 + T^{2} )^{2}$$
$59$ $$( 6400 + T^{2} )^{2}$$
$61$ $$( -36 + T )^{4}$$
$67$ $$( -14112 + T^{2} )^{2}$$
$71$ $$( 11664 + T^{2} )^{2}$$
$73$ $$( -56 + T )^{4}$$
$79$ $$( -648 + T^{2} )^{2}$$
$83$ $$( 5776 + T^{2} )^{2}$$
$89$ $$( 7938 + T^{2} )^{2}$$
$97$ $$( -104 + T )^{4}$$