# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.l (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: $$2^{6}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{5} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{7} +O(q^{10})$$ $$q + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{5} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{7} + ( -2 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{11} + ( 1 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{13} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{17} + ( -4 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{19} + ( 2 - 4 \zeta_{24}^{4} ) q^{23} + ( 2 - 4 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{25} + ( -5 \zeta_{24} - \zeta_{24}^{3} + 5 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{29} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} - \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{31} + ( 2 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{35} + ( 3 + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{37} + ( -5 \zeta_{24} - 5 \zeta_{24}^{3} - \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{41} + ( -4 \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{43} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{47} -5 q^{49} + ( -5 \zeta_{24} + 3 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{53} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{55} + ( -2 - 8 \zeta_{24}^{2} + 8 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{59} + ( -3 - 3 \zeta_{24}^{6} ) q^{61} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{65} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{67} -14 \zeta_{24}^{6} q^{71} + ( 4 - 8 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{73} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{77} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 7 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{79} + ( 10 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 10 \zeta_{24}^{6} ) q^{83} + ( -2 + 6 \zeta_{24}^{2} - 6 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{85} + ( -5 \zeta_{24} - 5 \zeta_{24}^{3} - 7 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{89} + ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{91} + ( -10 + 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{95} + ( -8 + 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 8q^{11} + 8q^{35} + 8q^{37} - 40q^{49} + 16q^{59} - 24q^{61} + 72q^{83} - 40q^{85} - 80q^{95} - 64q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
575.1
 0.258819 − 0.965926i 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 0 0 −1.93185 + 1.93185i 0 −1.41421 0 0 0
575.2 0 0 0 −0.517638 + 0.517638i 0 1.41421 0 0 0
575.3 0 0 0 0.517638 0.517638i 0 −1.41421 0 0 0
575.4 0 0 0 1.93185 1.93185i 0 1.41421 0 0 0
1727.1 0 0 0 −1.93185 1.93185i 0 −1.41421 0 0 0
1727.2 0 0 0 −0.517638 0.517638i 0 1.41421 0 0 0
1727.3 0 0 0 0.517638 + 0.517638i 0 −1.41421 0 0 0
1727.4 0 0 0 1.93185 + 1.93185i 0 1.41421 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1727.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner
16.e even 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.l.c yes 8
3.b odd 2 1 2304.2.l.g yes 8
4.b odd 2 1 2304.2.l.g yes 8
8.b even 2 1 2304.2.l.f yes 8
8.d odd 2 1 2304.2.l.b 8
12.b even 2 1 inner 2304.2.l.c yes 8
16.e even 4 1 inner 2304.2.l.c yes 8
16.e even 4 1 2304.2.l.f yes 8
16.f odd 4 1 2304.2.l.b 8
16.f odd 4 1 2304.2.l.g yes 8
24.f even 2 1 2304.2.l.f yes 8
24.h odd 2 1 2304.2.l.b 8
48.i odd 4 1 2304.2.l.b 8
48.i odd 4 1 2304.2.l.g yes 8
48.k even 4 1 inner 2304.2.l.c yes 8
48.k even 4 1 2304.2.l.f yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2304.2.l.b 8 8.d odd 2 1
2304.2.l.b 8 16.f odd 4 1
2304.2.l.b 8 24.h odd 2 1
2304.2.l.b 8 48.i odd 4 1
2304.2.l.c yes 8 1.a even 1 1 trivial
2304.2.l.c yes 8 12.b even 2 1 inner
2304.2.l.c yes 8 16.e even 4 1 inner
2304.2.l.c yes 8 48.k even 4 1 inner
2304.2.l.f yes 8 8.b even 2 1
2304.2.l.f yes 8 16.e even 4 1
2304.2.l.f yes 8 24.f even 2 1
2304.2.l.f yes 8 48.k even 4 1
2304.2.l.g yes 8 3.b odd 2 1
2304.2.l.g yes 8 4.b odd 2 1
2304.2.l.g yes 8 16.f odd 4 1
2304.2.l.g yes 8 48.i odd 4 1

## Hecke kernels

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

 $$T_{5}^{8} + 56 T_{5}^{4} + 16$$ $$T_{11}^{4} + 4 T_{11}^{3} + 8 T_{11}^{2} - 16 T_{11} + 16$$ $$T_{13}^{4} + 36$$ $$T_{37}^{4} - 4 T_{37}^{3} + 8 T_{37}^{2} + 88 T_{37} + 484$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 4 T^{4} - 474 T^{8} - 2500 T^{12} + 390625 T^{16}$$
$7$ $$( 1 + 12 T^{2} + 49 T^{4} )^{4}$$
$11$ $$( 1 + 4 T + 8 T^{2} + 28 T^{3} + 82 T^{4} + 308 T^{5} + 968 T^{6} + 5324 T^{7} + 14641 T^{8} )^{2}$$
$13$ $$( 1 + 62 T^{4} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 40 T^{2} + 786 T^{4} - 11560 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$1 + 292 T^{4} - 25242 T^{8} + 38053732 T^{12} + 16983563041 T^{16}$$
$23$ $$( 1 - 34 T^{2} + 529 T^{4} )^{4}$$
$29$ $$1 - 964 T^{4} + 566886 T^{8} - 681818884 T^{12} + 500246412961 T^{16}$$
$31$ $$( 1 - 40 T^{2} + 594 T^{4} - 38440 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 4 T + 8 T^{2} - 60 T^{3} - 34 T^{4} - 2220 T^{5} + 10952 T^{6} - 202612 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 + 40 T^{2} + 2034 T^{4} + 67240 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$1 - 3548 T^{4} + 6433446 T^{8} - 12129905948 T^{12} + 11688200277601 T^{16}$$
$47$ $$( 1 - 14 T^{2} + 2209 T^{4} )^{4}$$
$53$ $$1 - 4868 T^{4} + 16478118 T^{8} - 38410861508 T^{12} + 62259690411361 T^{16}$$
$59$ $$( 1 - 8 T + 32 T^{2} + 232 T^{3} - 6062 T^{4} + 13688 T^{5} + 111392 T^{6} - 1643032 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 6 T + 18 T^{2} + 366 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$1 + 10276 T^{4} + 56007654 T^{8} + 207072919396 T^{12} + 406067677556641 T^{16}$$
$71$ $$( 1 + 54 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 188 T^{2} + 18726 T^{4} - 1001852 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 168 T^{2} + 14738 T^{4} - 1048488 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 - 36 T + 648 T^{2} - 8604 T^{3} + 89906 T^{4} - 714132 T^{5} + 4464072 T^{6} - 20584332 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 80 T^{2} + 15714 T^{4} - 633680 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 16 T + 150 T^{2} + 1552 T^{3} + 9409 T^{4} )^{4}$$
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