# Properties

 Label 2304.3.h.i Level $2304$ Weight $3$ Character orbit 2304.h Analytic conductor $62.779$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## 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: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{41}]$$ Coefficient ring index: $$2^{14}$$ 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24} + 4 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{5} + ( -2 + 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{7} +O(q^{10})$$ $$q + ( -\zeta_{24} + 4 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{5} + ( -2 + 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{7} + ( -2 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{11} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 8 \zeta_{24}^{6} + 12 \zeta_{24}^{7} ) q^{13} + ( -12 + \zeta_{24} - \zeta_{24}^{3} + 24 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{17} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{19} + ( -10 \zeta_{24} + 10 \zeta_{24}^{3} + 10 \zeta_{24}^{5} ) q^{23} + ( -11 - 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{25} + ( 3 \zeta_{24} + 28 \zeta_{24}^{2} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 14 \zeta_{24}^{6} ) q^{29} + ( -30 + 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{31} + ( 26 \zeta_{24} - 24 \zeta_{24}^{2} + 26 \zeta_{24}^{3} - 26 \zeta_{24}^{5} + 12 \zeta_{24}^{6} ) q^{35} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} + 22 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{37} + ( -20 + 21 \zeta_{24} - 21 \zeta_{24}^{3} + 40 \zeta_{24}^{4} - 21 \zeta_{24}^{5} ) q^{41} + ( 16 \zeta_{24} - 16 \zeta_{24}^{3} + 16 \zeta_{24}^{5} + 36 \zeta_{24}^{6} + 32 \zeta_{24}^{7} ) q^{43} + ( -16 + 10 \zeta_{24} - 10 \zeta_{24}^{3} + 32 \zeta_{24}^{4} - 10 \zeta_{24}^{5} ) q^{47} + ( 51 - 16 \zeta_{24} - 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} + 32 \zeta_{24}^{7} ) q^{49} + ( -27 \zeta_{24} + 20 \zeta_{24}^{2} - 27 \zeta_{24}^{3} + 27 \zeta_{24}^{5} - 10 \zeta_{24}^{6} ) q^{53} + ( 28 - 8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 16 \zeta_{24}^{7} ) q^{55} + ( 20 \zeta_{24} - 96 \zeta_{24}^{2} + 20 \zeta_{24}^{3} - 20 \zeta_{24}^{5} + 48 \zeta_{24}^{6} ) q^{59} + ( -16 \zeta_{24} + 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} - 50 \zeta_{24}^{6} - 32 \zeta_{24}^{7} ) q^{61} + ( 28 - 44 \zeta_{24} + 44 \zeta_{24}^{3} - 56 \zeta_{24}^{4} + 44 \zeta_{24}^{5} ) q^{65} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 28 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{67} + ( -24 - 54 \zeta_{24} + 54 \zeta_{24}^{3} + 48 \zeta_{24}^{4} + 54 \zeta_{24}^{5} ) q^{71} + ( -68 + 8 \zeta_{24} + 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} - 16 \zeta_{24}^{7} ) q^{73} + ( 52 \zeta_{24} - 48 \zeta_{24}^{2} + 52 \zeta_{24}^{3} - 52 \zeta_{24}^{5} + 24 \zeta_{24}^{6} ) q^{77} + ( -122 + 12 \zeta_{24} + 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} - 24 \zeta_{24}^{7} ) q^{79} + ( 78 \zeta_{24} + 56 \zeta_{24}^{2} + 78 \zeta_{24}^{3} - 78 \zeta_{24}^{5} - 28 \zeta_{24}^{6} ) q^{83} + ( -14 \zeta_{24} + 14 \zeta_{24}^{3} - 14 \zeta_{24}^{5} + 74 \zeta_{24}^{6} - 28 \zeta_{24}^{7} ) q^{85} + ( 56 + 17 \zeta_{24} - 17 \zeta_{24}^{3} - 112 \zeta_{24}^{4} - 17 \zeta_{24}^{5} ) q^{89} + ( -44 \zeta_{24} + 44 \zeta_{24}^{3} - 44 \zeta_{24}^{5} + 160 \zeta_{24}^{6} - 88 \zeta_{24}^{7} ) q^{91} + ( 24 - 32 \zeta_{24} + 32 \zeta_{24}^{3} - 48 \zeta_{24}^{4} + 32 \zeta_{24}^{5} ) q^{95} + ( 40 + 52 \zeta_{24} + 52 \zeta_{24}^{3} + 52 \zeta_{24}^{5} - 104 \zeta_{24}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 16q^{7} + O(q^{10})$$ $$8q - 16q^{7} - 88q^{25} - 240q^{31} + 408q^{49} + 224q^{55} - 544q^{73} - 976q^{79} + 320q^{97} + O(q^{100})$$

## 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
 −0.258819 − 0.965926i −0.258819 + 0.965926i 0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 − 0.258819i 0.965926 + 0.258819i −0.965926 − 0.258819i −0.965926 + 0.258819i
0 0 0 −4.87832 0 −11.7980 0 0 0
2177.2 0 0 0 −4.87832 0 −11.7980 0 0 0
2177.3 0 0 0 −2.04989 0 7.79796 0 0 0
2177.4 0 0 0 −2.04989 0 7.79796 0 0 0
2177.5 0 0 0 2.04989 0 7.79796 0 0 0
2177.6 0 0 0 2.04989 0 7.79796 0 0 0
2177.7 0 0 0 4.87832 0 −11.7980 0 0 0
2177.8 0 0 0 4.87832 0 −11.7980 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2177.8 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.i 8
3.b odd 2 1 inner 2304.3.h.i 8
4.b odd 2 1 2304.3.h.k 8
8.b even 2 1 inner 2304.3.h.i 8
8.d odd 2 1 2304.3.h.k 8
12.b even 2 1 2304.3.h.k 8
16.e even 4 1 1152.3.e.f yes 4
16.e even 4 1 1152.3.e.h yes 4
16.f odd 4 1 1152.3.e.b 4
16.f odd 4 1 1152.3.e.d yes 4
24.f even 2 1 2304.3.h.k 8
24.h odd 2 1 inner 2304.3.h.i 8
48.i odd 4 1 1152.3.e.f yes 4
48.i odd 4 1 1152.3.e.h yes 4
48.k even 4 1 1152.3.e.b 4
48.k even 4 1 1152.3.e.d yes 4

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

## Hecke kernels

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

 $$T_{5}^{4} - 28 T_{5}^{2} + 100$$ $$T_{7}^{2} + 4 T_{7} - 92$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 100 - 28 T^{2} + T^{4} )^{2}$$
$7$ $$( -92 + 4 T + T^{2} )^{4}$$
$11$ $$( 1600 - 112 T^{2} + T^{4} )^{2}$$
$13$ $$( 23104 + 560 T^{2} + T^{4} )^{2}$$
$17$ $$( 184900 + 868 T^{2} + T^{4} )^{2}$$
$19$ $$( 1024 + 320 T^{2} + T^{4} )^{2}$$
$23$ $$( 200 + T^{2} )^{4}$$
$29$ $$( 324900 - 1212 T^{2} + T^{4} )^{2}$$
$31$ $$( 804 + 60 T + T^{2} )^{4}$$
$37$ $$( 10000 + 1736 T^{2} + T^{4} )^{2}$$
$41$ $$( 101124 + 4164 T^{2} + T^{4} )^{2}$$
$43$ $$( 57600 + 5664 T^{2} + T^{4} )^{2}$$
$47$ $$( 322624 + 1936 T^{2} + T^{4} )^{2}$$
$53$ $$( 1340964 - 3516 T^{2} + T^{4} )^{2}$$
$59$ $$( 37356544 - 15424 T^{2} + T^{4} )^{2}$$
$61$ $$( 929296 + 8072 T^{2} + T^{4} )^{2}$$
$67$ $$( 473344 + 1760 T^{2} + T^{4} )^{2}$$
$71$ $$( 16842816 + 15120 T^{2} + T^{4} )^{2}$$
$73$ $$( 4240 + 136 T + T^{2} )^{4}$$
$79$ $$( 14020 + 244 T + T^{2} )^{4}$$
$83$ $$( 96353856 - 29040 T^{2} + T^{4} )^{2}$$
$89$ $$( 77968900 + 19972 T^{2} + T^{4} )^{2}$$
$97$ $$( -14624 - 80 T + T^{2} )^{4}$$