# Properties

 Label 2880.2.bj.a Level $2880$ Weight $2$ Character orbit 2880.bj Analytic conductor $22.997$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2880 = 2^{6} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2880.bj (of order $$4$$, degree $$2$$, 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: $$2^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{24}^{3} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{7} +O(q^{10})$$ $$q + ( -1 - \zeta_{24}^{3} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{7} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{11} + ( -1 + 2 \zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{13} + ( 2 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{17} + ( -2 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{19} + ( -2 + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{23} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{25} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{29} + ( 2 \zeta_{24} - 8 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{31} + ( 2 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} ) q^{35} + ( -1 + 2 \zeta_{24}^{2} + 6 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{37} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{41} + ( 2 - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{43} + ( -4 - 2 \zeta_{24} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{47} + 3 \zeta_{24}^{6} q^{49} + ( 2 - 4 \zeta_{24} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{53} + ( 2 - 6 \zeta_{24} - 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{55} + ( -2 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} ) q^{59} + ( -6 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 12 \zeta_{24}^{4} + 2 \zeta_{24}^{5} ) q^{61} + ( -2 + \zeta_{24} + 4 \zeta_{24}^{2} - 5 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{65} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 8 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{71} + ( -3 - 4 \zeta_{24} - 4 \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{73} + ( -4 \zeta_{24}^{3} + 8 \zeta_{24}^{7} ) q^{77} + ( -10 \zeta_{24} + 10 \zeta_{24}^{3} + 10 \zeta_{24}^{5} ) q^{79} + ( 2 - 4 \zeta_{24}^{2} - 8 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{83} + ( -4 + 8 \zeta_{24} - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 8 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{85} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{89} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{91} + ( -4 - 4 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 8 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{95} + ( -3 - 4 \zeta_{24}^{3} + 3 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{5} + O(q^{10})$$ $$8q - 8q^{5} - 16q^{19} - 16q^{23} + 16q^{43} - 32q^{47} + 16q^{53} - 24q^{73} - 32q^{95} - 24q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
737.1
 0.965926 − 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 − 0.258819i
0 0 0 −2.22474 0.224745i 0 −1.41421 1.41421i 0 0 0
737.2 0 0 0 −2.22474 0.224745i 0 1.41421 + 1.41421i 0 0 0
737.3 0 0 0 0.224745 + 2.22474i 0 −1.41421 1.41421i 0 0 0
737.4 0 0 0 0.224745 + 2.22474i 0 1.41421 + 1.41421i 0 0 0
1313.1 0 0 0 −2.22474 + 0.224745i 0 −1.41421 + 1.41421i 0 0 0
1313.2 0 0 0 −2.22474 + 0.224745i 0 1.41421 1.41421i 0 0 0
1313.3 0 0 0 0.224745 2.22474i 0 −1.41421 + 1.41421i 0 0 0
1313.4 0 0 0 0.224745 2.22474i 0 1.41421 1.41421i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1313.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner
24.f even 2 1 inner
120.w even 4 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{7}^{4} + 16$$ $$T_{19}^{2} + 4 T_{19} - 20$$ $$T_{23}^{4} + 8 T_{23}^{3} + 32 T_{23}^{2} - 32 T_{23} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 25 + 20 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$7$ $$( 16 + T^{4} )^{2}$$
$11$ $$( -12 + T^{2} )^{4}$$
$13$ $$16 + 392 T^{4} + T^{8}$$
$17$ $$160000 + 2336 T^{4} + T^{8}$$
$19$ $$( -20 + 4 T + T^{2} )^{4}$$
$23$ $$( 16 - 32 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$29$ $$( 900 + 84 T^{2} + T^{4} )^{2}$$
$31$ $$( 1600 - 112 T^{2} + T^{4} )^{2}$$
$37$ $$810000 + 5256 T^{4} + T^{8}$$
$41$ $$( 2 + T^{2} )^{4}$$
$43$ $$( 1600 + 320 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$47$ $$( 400 + 320 T + 128 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$53$ $$( 1600 + 320 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$59$ $$( 16 + 40 T^{2} + T^{4} )^{2}$$
$61$ $$( 10000 + 232 T^{2} + T^{4} )^{2}$$
$67$ $$T^{8}$$
$71$ $$( 1600 + 176 T^{2} + T^{4} )^{2}$$
$73$ $$( 900 - 360 T + 72 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$79$ $$( 200 + T^{2} )^{4}$$
$83$ $$2560000 + 27776 T^{4} + T^{8}$$
$89$ $$( -18 + T^{2} )^{4}$$
$97$ $$( 900 - 360 T + 72 T^{2} + 12 T^{3} + T^{4} )^{2}$$