# Properties

 Label 2304.2.k.e Level $2304$ Weight $2$ Character orbit 2304.k Analytic conductor $18.398$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.3975326257$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 768) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} +O(q^{10})$$ $$q + ( -2 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{11} + ( 3 + 3 \zeta_{24}^{6} ) q^{13} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{17} + ( -2 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{19} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{23} + ( 4 - 8 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{25} + ( -2 + 6 \zeta_{24}^{2} + 6 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{29} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{31} + ( 2 \zeta_{24} + 6 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{35} + ( -1 + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{37} + ( -2 + 4 \zeta_{24}^{4} - 8 \zeta_{24}^{6} ) q^{41} + ( -2 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{43} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{47} + q^{49} + ( -4 + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{53} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{55} + ( 8 \zeta_{24}^{3} - 16 \zeta_{24}^{7} ) q^{59} + ( -5 + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} - 5 \zeta_{24}^{6} ) q^{61} + ( -6 - 12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{65} + ( 4 \zeta_{24} + 8 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{67} + ( -10 \zeta_{24} + 10 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{71} + 4 \zeta_{24}^{6} q^{73} + ( -8 + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} - 8 \zeta_{24}^{6} ) q^{77} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{79} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{83} + ( 8 + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 8 \zeta_{24}^{6} ) q^{85} + ( 4 - 8 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{89} + ( 6 \zeta_{24}^{3} - 12 \zeta_{24}^{7} ) q^{91} + ( 16 \zeta_{24} + 16 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{95} + ( 8 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{5} + O(q^{10})$$ $$8q - 8q^{5} + 24q^{13} + 8q^{29} - 24q^{37} + 8q^{49} - 40q^{53} - 24q^{61} - 48q^{65} - 48q^{77} + 48q^{85} + 64q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{24}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 0.965926 + 0.258819i −0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i
0 0 0 −2.73205 + 2.73205i 0 2.44949i 0 0 0
577.2 0 0 0 −2.73205 + 2.73205i 0 2.44949i 0 0 0
577.3 0 0 0 0.732051 0.732051i 0 2.44949i 0 0 0
577.4 0 0 0 0.732051 0.732051i 0 2.44949i 0 0 0
1729.1 0 0 0 −2.73205 2.73205i 0 2.44949i 0 0 0
1729.2 0 0 0 −2.73205 2.73205i 0 2.44949i 0 0 0
1729.3 0 0 0 0.732051 + 0.732051i 0 2.44949i 0 0 0
1729.4 0 0 0 0.732051 + 0.732051i 0 2.44949i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1729.4 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
16.e even 4 1 inner
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.k.e 8
3.b odd 2 1 768.2.j.f yes 8
4.b odd 2 1 inner 2304.2.k.e 8
8.b even 2 1 2304.2.k.l 8
8.d odd 2 1 2304.2.k.l 8
12.b even 2 1 768.2.j.f yes 8
16.e even 4 1 inner 2304.2.k.e 8
16.e even 4 1 2304.2.k.l 8
16.f odd 4 1 inner 2304.2.k.e 8
16.f odd 4 1 2304.2.k.l 8
24.f even 2 1 768.2.j.e 8
24.h odd 2 1 768.2.j.e 8
32.g even 8 1 9216.2.a.bc 4
32.g even 8 1 9216.2.a.bi 4
32.h odd 8 1 9216.2.a.bc 4
32.h odd 8 1 9216.2.a.bi 4
48.i odd 4 1 768.2.j.e 8
48.i odd 4 1 768.2.j.f yes 8
48.k even 4 1 768.2.j.e 8
48.k even 4 1 768.2.j.f yes 8
96.o even 8 1 3072.2.a.l 4
96.o even 8 1 3072.2.a.r 4
96.o even 8 2 3072.2.d.h 8
96.p odd 8 1 3072.2.a.l 4
96.p odd 8 1 3072.2.a.r 4
96.p odd 8 2 3072.2.d.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.e 8 24.f even 2 1
768.2.j.e 8 24.h odd 2 1
768.2.j.e 8 48.i odd 4 1
768.2.j.e 8 48.k even 4 1
768.2.j.f yes 8 3.b odd 2 1
768.2.j.f yes 8 12.b even 2 1
768.2.j.f yes 8 48.i odd 4 1
768.2.j.f yes 8 48.k even 4 1
2304.2.k.e 8 1.a even 1 1 trivial
2304.2.k.e 8 4.b odd 2 1 inner
2304.2.k.e 8 16.e even 4 1 inner
2304.2.k.e 8 16.f odd 4 1 inner
2304.2.k.l 8 8.b even 2 1
2304.2.k.l 8 8.d odd 2 1
2304.2.k.l 8 16.e even 4 1
2304.2.k.l 8 16.f odd 4 1
3072.2.a.l 4 96.o even 8 1
3072.2.a.l 4 96.p odd 8 1
3072.2.a.r 4 96.o even 8 1
3072.2.a.r 4 96.p odd 8 1
3072.2.d.h 8 96.o even 8 2
3072.2.d.h 8 96.p odd 8 2
9216.2.a.bc 4 32.g even 8 1
9216.2.a.bc 4 32.h odd 8 1
9216.2.a.bi 4 32.g even 8 1
9216.2.a.bi 4 32.h odd 8 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5}^{4} + 4 T_{5}^{3} + 8 T_{5}^{2} - 16 T_{5} + 16$$ $$T_{7}^{2} + 6$$ $$T_{13}^{2} - 6 T_{13} + 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 4 T + 8 T^{2} + 4 T^{3} - 14 T^{4} + 20 T^{5} + 200 T^{6} + 500 T^{7} + 625 T^{8} )^{2}$$
$7$ $$( 1 - 8 T^{2} + 49 T^{4} )^{4}$$
$11$ $$1 - 28 T^{4} + 1830 T^{8} - 409948 T^{12} + 214358881 T^{16}$$
$13$ $$( 1 - 6 T + 18 T^{2} - 78 T^{3} + 169 T^{4} )^{4}$$
$17$ $$( 1 + 22 T^{2} + 289 T^{4} )^{4}$$
$19$ $$1 + 292 T^{4} - 25242 T^{8} + 38053732 T^{12} + 16983563041 T^{16}$$
$23$ $$( 1 - 38 T^{2} + 529 T^{4} )^{4}$$
$29$ $$( 1 - 4 T + 8 T^{2} + 92 T^{3} - 1646 T^{4} + 2668 T^{5} + 6728 T^{6} - 97556 T^{7} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 8 T^{2} + 961 T^{4} )^{4}$$
$37$ $$( 1 + 12 T + 72 T^{2} + 372 T^{3} + 1886 T^{4} + 13764 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 - 12 T^{2} + 326 T^{4} - 20172 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 1778 T^{4} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 86 T^{2} + 2209 T^{4} )^{4}$$
$53$ $$( 1 + 20 T + 200 T^{2} + 1940 T^{3} + 16882 T^{4} + 102820 T^{5} + 561800 T^{6} + 2977540 T^{7} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 1486 T^{4} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 12 T + 72 T^{2} + 660 T^{3} + 6014 T^{4} + 40260 T^{5} + 267912 T^{6} + 2723772 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$1 + 7588 T^{4} + 30907110 T^{8} + 152906706148 T^{12} + 406067677556641 T^{16}$$
$71$ $$( 1 + 20 T^{2} - 2106 T^{4} + 100820 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 130 T^{2} + 5329 T^{4} )^{4}$$
$79$ $$( 1 + 152 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$1 + 17828 T^{4} + 157096038 T^{8} + 846086946788 T^{12} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 252 T^{2} + 30950 T^{4} - 1996092 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 16 T + 210 T^{2} - 1552 T^{3} + 9409 T^{4} )^{4}$$