# Properties

 Label 2304.2.k.h Level $2304$ Weight $2$ Character orbit 2304.k 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.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: $$2^{10}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{7} +O(q^{10})$$ $$q + ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{7} + ( 2 - 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{11} + ( -1 + \zeta_{24}^{6} ) q^{13} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{17} + ( 6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{19} + ( -4 + 8 \zeta_{24}^{4} ) q^{23} + 7 \zeta_{24}^{6} q^{25} + ( -2 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{29} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{31} + ( 2 + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{35} + ( 7 + 7 \zeta_{24}^{6} ) q^{37} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{41} -6 \zeta_{24}^{3} q^{43} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{47} + 5 q^{49} + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{53} + ( -12 \zeta_{24} + 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} ) q^{55} + ( -4 + 8 \zeta_{24}^{2} + 8 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{59} + ( -5 + 5 \zeta_{24}^{6} ) q^{61} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{65} + ( 8 \zeta_{24} - 8 \zeta_{24}^{5} ) q^{67} + ( 8 - 16 \zeta_{24}^{4} ) q^{71} + 12 \zeta_{24}^{6} q^{73} + ( 4 \zeta_{24}^{3} - 8 \zeta_{24}^{7} ) q^{77} + ( -11 \zeta_{24} - 11 \zeta_{24}^{3} + 11 \zeta_{24}^{5} ) q^{79} + ( -6 - 12 \zeta_{24}^{2} + 12 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{83} + ( 12 + 12 \zeta_{24}^{6} ) q^{85} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{89} -2 \zeta_{24}^{3} q^{91} + ( -24 \zeta_{24}^{2} + 12 \zeta_{24}^{6} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 8q^{13} + 56q^{37} + 40q^{49} - 40q^{61} + 96q^{85} + O(q^{100})$$

## Character values

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

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

## 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 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 + 0.258819i 0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i −0.258819 − 0.965926i
0 0 0 −2.44949 + 2.44949i 0 1.41421i 0 0 0
577.2 0 0 0 −2.44949 + 2.44949i 0 1.41421i 0 0 0
577.3 0 0 0 2.44949 2.44949i 0 1.41421i 0 0 0
577.4 0 0 0 2.44949 2.44949i 0 1.41421i 0 0 0
1729.1 0 0 0 −2.44949 2.44949i 0 1.41421i 0 0 0
1729.2 0 0 0 −2.44949 2.44949i 0 1.41421i 0 0 0
1729.3 0 0 0 2.44949 + 2.44949i 0 1.41421i 0 0 0
1729.4 0 0 0 2.44949 + 2.44949i 0 1.41421i 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
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner
48.i odd 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.k.h 8
3.b odd 2 1 inner 2304.2.k.h 8
4.b odd 2 1 inner 2304.2.k.h 8
8.b even 2 1 2304.2.k.i yes 8
8.d odd 2 1 2304.2.k.i yes 8
12.b even 2 1 inner 2304.2.k.h 8
16.e even 4 1 inner 2304.2.k.h 8
16.e even 4 1 2304.2.k.i yes 8
16.f odd 4 1 inner 2304.2.k.h 8
16.f odd 4 1 2304.2.k.i yes 8
24.f even 2 1 2304.2.k.i yes 8
24.h odd 2 1 2304.2.k.i yes 8
32.g even 8 1 9216.2.a.be 4
32.g even 8 1 9216.2.a.bh 4
32.h odd 8 1 9216.2.a.be 4
32.h odd 8 1 9216.2.a.bh 4
48.i odd 4 1 inner 2304.2.k.h 8
48.i odd 4 1 2304.2.k.i yes 8
48.k even 4 1 inner 2304.2.k.h 8
48.k even 4 1 2304.2.k.i yes 8
96.o even 8 1 9216.2.a.be 4
96.o even 8 1 9216.2.a.bh 4
96.p odd 8 1 9216.2.a.be 4
96.p odd 8 1 9216.2.a.bh 4

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

## Hecke kernels

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

 $$T_{5}^{4} + 144$$ $$T_{7}^{2} + 2$$ $$T_{13}^{2} + 2 T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 144 + T^{4} )^{2}$$
$7$ $$( 2 + T^{2} )^{4}$$
$11$ $$( 576 + T^{4} )^{2}$$
$13$ $$( 2 + 2 T + T^{2} )^{4}$$
$17$ $$( -24 + T^{2} )^{4}$$
$19$ $$( 1296 + T^{4} )^{2}$$
$23$ $$( 48 + T^{2} )^{4}$$
$29$ $$( 144 + T^{4} )^{2}$$
$31$ $$( -2 + T^{2} )^{4}$$
$37$ $$( 98 - 14 T + T^{2} )^{4}$$
$41$ $$( 24 + T^{2} )^{4}$$
$43$ $$( 1296 + T^{4} )^{2}$$
$47$ $$( -48 + T^{2} )^{4}$$
$53$ $$( 144 + T^{4} )^{2}$$
$59$ $$( 9216 + T^{4} )^{2}$$
$61$ $$( 50 + 10 T + T^{2} )^{4}$$
$67$ $$( 4096 + T^{4} )^{2}$$
$71$ $$( 192 + T^{2} )^{4}$$
$73$ $$( 144 + T^{2} )^{4}$$
$79$ $$( -242 + T^{2} )^{4}$$
$83$ $$( 46656 + T^{4} )^{2}$$
$89$ $$( 96 + T^{2} )^{4}$$
$97$ $$T^{8}$$