# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.3975326257$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{12}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 576) 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{5} -2 \zeta_{24}^{6} q^{7} +O(q^{10})$$ $$q + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{5} -2 \zeta_{24}^{6} q^{7} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{11} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{13} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{17} + ( 4 - 8 \zeta_{24}^{4} ) q^{19} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{23} - q^{25} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{29} -2 \zeta_{24}^{6} q^{31} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{35} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{37} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{41} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{47} + 3 q^{49} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{53} -12 \zeta_{24}^{6} q^{55} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{61} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{65} + ( -4 + 8 \zeta_{24}^{4} ) q^{67} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{71} -4 q^{73} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{77} -14 \zeta_{24}^{6} q^{79} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{83} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{85} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{89} + ( -4 + 8 \zeta_{24}^{4} ) q^{91} + ( 12 \zeta_{24} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} ) q^{95} -16 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 8q^{25} + 24q^{49} - 32q^{73} - 128q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2303.1
 −0.965926 − 0.258819i −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 − 0.965926i 0.965926 + 0.258819i 0.258819 + 0.965926i −0.965926 + 0.258819i −0.258819 + 0.965926i
0 0 0 2.44949i 0 2.00000i 0 0 0
2303.2 0 0 0 2.44949i 0 2.00000i 0 0 0
2303.3 0 0 0 2.44949i 0 2.00000i 0 0 0
2303.4 0 0 0 2.44949i 0 2.00000i 0 0 0
2303.5 0 0 0 2.44949i 0 2.00000i 0 0 0
2303.6 0 0 0 2.44949i 0 2.00000i 0 0 0
2303.7 0 0 0 2.44949i 0 2.00000i 0 0 0
2303.8 0 0 0 2.44949i 0 2.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2303.8 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f 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 2304.2.c.j 8
3.b odd 2 1 inner 2304.2.c.j 8
4.b odd 2 1 inner 2304.2.c.j 8
8.b even 2 1 inner 2304.2.c.j 8
8.d odd 2 1 inner 2304.2.c.j 8
12.b even 2 1 inner 2304.2.c.j 8
16.e even 4 2 576.2.f.a 8
16.f odd 4 2 576.2.f.a 8
24.f even 2 1 inner 2304.2.c.j 8
24.h odd 2 1 inner 2304.2.c.j 8
48.i odd 4 2 576.2.f.a 8
48.k even 4 2 576.2.f.a 8

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

## Hecke kernels

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

 $$T_{5}^{2} + 6$$ $$T_{7}^{2} + 4$$ $$T_{11}^{2} - 24$$ $$T_{13}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 4 T^{2} + 25 T^{4} )^{4}$$
$7$ $$( 1 - 10 T^{2} + 49 T^{4} )^{4}$$
$11$ $$( 1 - 2 T^{2} + 121 T^{4} )^{4}$$
$13$ $$( 1 + 14 T^{2} + 169 T^{4} )^{4}$$
$17$ $$( 1 - 16 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 + 10 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 26 T^{2} + 529 T^{4} )^{4}$$
$29$ $$( 1 - 4 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - 58 T^{2} + 961 T^{4} )^{4}$$
$37$ $$( 1 + 26 T^{2} + 1369 T^{4} )^{4}$$
$41$ $$( 1 - 64 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 - 43 T^{2} )^{8}$$
$47$ $$( 1 + 22 T^{2} + 2209 T^{4} )^{4}$$
$53$ $$( 1 - 100 T^{2} + 2809 T^{4} )^{4}$$
$59$ $$( 1 + 59 T^{2} )^{8}$$
$61$ $$( 1 + 74 T^{2} + 3721 T^{4} )^{4}$$
$67$ $$( 1 - 86 T^{2} + 4489 T^{4} )^{4}$$
$71$ $$( 1 + 70 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 + 4 T + 73 T^{2} )^{8}$$
$79$ $$( 1 + 38 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 142 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 - 160 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 + 16 T + 97 T^{2} )^{8}$$