# Properties

 Label 2880.2.w.n Level $2880$ Weight $2$ Character orbit 2880.w Analytic conductor $22.997$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2880 = 2^{6} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2880.w (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.9969157821$$ 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: no (minimal twist has level 1440) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{5} + ( 2 + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{5} + ( 2 + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{7} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{11} + ( -1 - \zeta_{24}^{6} ) q^{13} -4 \zeta_{24}^{3} q^{17} + ( 4 - 8 \zeta_{24}^{4} ) q^{19} + ( 4 - 3 \zeta_{24}^{6} ) q^{25} + ( 5 \zeta_{24} + 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{29} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{31} + ( -2 \zeta_{24} + 6 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{35} + ( -1 + \zeta_{24}^{6} ) q^{37} + ( 7 \zeta_{24} - 7 \zeta_{24}^{3} - 7 \zeta_{24}^{5} ) q^{41} + ( -4 + 8 \zeta_{24}^{2} + 8 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{43} + ( 4 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{47} -17 \zeta_{24}^{6} q^{49} + ( 4 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{53} + ( -6 + 4 \zeta_{24}^{2} + 12 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{55} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{59} + ( -\zeta_{24} - 3 \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{65} + ( -4 - 8 \zeta_{24}^{2} + 8 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{67} + ( -7 - 7 \zeta_{24}^{6} ) q^{73} + 24 \zeta_{24}^{3} q^{77} + ( 4 - 8 \zeta_{24}^{4} ) q^{79} + ( -4 \zeta_{24}^{3} + 8 \zeta_{24}^{7} ) q^{83} + ( -8 - 4 \zeta_{24}^{6} ) q^{85} + ( 7 \zeta_{24} + 7 \zeta_{24}^{3} - 7 \zeta_{24}^{5} ) q^{89} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{91} + ( -8 \zeta_{24} + 4 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{95} + ( 5 - 5 \zeta_{24}^{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 8q^{13} + 32q^{25} - 8q^{37} - 56q^{73} - 64q^{85} + 40q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$641$$ $$901$$ $$2431$$ $$\chi(n)$$ $$-\zeta_{24}^{2}$$ $$-1$$ $$1$$ $$1$$

## 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 −0.965926 + 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 0 0 −2.12132 0.707107i 0 −3.46410 3.46410i 0 0 0
2177.2 0 0 0 −2.12132 0.707107i 0 3.46410 + 3.46410i 0 0 0
2177.3 0 0 0 2.12132 + 0.707107i 0 −3.46410 3.46410i 0 0 0
2177.4 0 0 0 2.12132 + 0.707107i 0 3.46410 + 3.46410i 0 0 0
2753.1 0 0 0 −2.12132 + 0.707107i 0 −3.46410 + 3.46410i 0 0 0
2753.2 0 0 0 −2.12132 + 0.707107i 0 3.46410 3.46410i 0 0 0
2753.3 0 0 0 2.12132 0.707107i 0 −3.46410 + 3.46410i 0 0 0
2753.4 0 0 0 2.12132 0.707107i 0 3.46410 3.46410i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2753.4 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
5.c odd 4 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
60.l odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.w.n 8
3.b odd 2 1 inner 2880.2.w.n 8
4.b odd 2 1 inner 2880.2.w.n 8
5.c odd 4 1 inner 2880.2.w.n 8
8.b even 2 1 1440.2.w.e 8
8.d odd 2 1 1440.2.w.e 8
12.b even 2 1 inner 2880.2.w.n 8
15.e even 4 1 inner 2880.2.w.n 8
20.e even 4 1 inner 2880.2.w.n 8
24.f even 2 1 1440.2.w.e 8
24.h odd 2 1 1440.2.w.e 8
40.i odd 4 1 1440.2.w.e 8
40.k even 4 1 1440.2.w.e 8
60.l odd 4 1 inner 2880.2.w.n 8
120.q odd 4 1 1440.2.w.e 8
120.w even 4 1 1440.2.w.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.w.e 8 8.b even 2 1
1440.2.w.e 8 8.d odd 2 1
1440.2.w.e 8 24.f even 2 1
1440.2.w.e 8 24.h odd 2 1
1440.2.w.e 8 40.i odd 4 1
1440.2.w.e 8 40.k even 4 1
1440.2.w.e 8 120.q odd 4 1
1440.2.w.e 8 120.w even 4 1
2880.2.w.n 8 1.a even 1 1 trivial
2880.2.w.n 8 3.b odd 2 1 inner
2880.2.w.n 8 4.b odd 2 1 inner
2880.2.w.n 8 5.c odd 4 1 inner
2880.2.w.n 8 12.b even 2 1 inner
2880.2.w.n 8 15.e even 4 1 inner
2880.2.w.n 8 20.e even 4 1 inner
2880.2.w.n 8 60.l odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2880, [\chi])$$:

 $$T_{7}^{4} + 576$$ $$T_{11}^{2} + 24$$ $$T_{13}^{2} + 2 T_{13} + 2$$ $$T_{31}^{2} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 25 - 8 T^{2} + T^{4} )^{2}$$
$7$ $$( 576 + T^{4} )^{2}$$
$11$ $$( 24 + T^{2} )^{4}$$
$13$ $$( 2 + 2 T + T^{2} )^{4}$$
$17$ $$( 256 + T^{4} )^{2}$$
$19$ $$( 48 + T^{2} )^{4}$$
$23$ $$T^{8}$$
$29$ $$( -50 + T^{2} )^{4}$$
$31$ $$( -48 + T^{2} )^{4}$$
$37$ $$( 2 + 2 T + T^{2} )^{4}$$
$41$ $$( 98 + T^{2} )^{4}$$
$43$ $$( 9216 + T^{4} )^{2}$$
$47$ $$( 2304 + T^{4} )^{2}$$
$53$ $$( 256 + T^{4} )^{2}$$
$59$ $$( -24 + T^{2} )^{4}$$
$61$ $$T^{8}$$
$67$ $$( 9216 + T^{4} )^{2}$$
$71$ $$T^{8}$$
$73$ $$( 98 + 14 T + T^{2} )^{4}$$
$79$ $$( 48 + T^{2} )^{4}$$
$83$ $$( 2304 + T^{4} )^{2}$$
$89$ $$( -98 + T^{2} )^{4}$$
$97$ $$( 50 - 10 T + T^{2} )^{4}$$