# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$2880 = 2^{6} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2880.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.9969157821$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 320) 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{12}^{3} q^{5} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} +O(q^{10})$$ $$q -\zeta_{12}^{3} q^{5} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + ( -2 + 4 \zeta_{12}^{2} ) q^{11} + ( 2 - 4 \zeta_{12}^{2} ) q^{13} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + 2 \zeta_{12}^{3} q^{19} + ( 3 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{23} - q^{25} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{35} -6 \zeta_{12}^{3} q^{37} + ( -6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{41} + ( -3 + 6 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{43} + ( -3 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{47} + ( 5 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{49} + ( -6 + 12 \zeta_{12}^{2} ) q^{53} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{55} + 6 \zeta_{12}^{3} q^{59} + ( -4 + 8 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{61} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{65} + ( -3 + 6 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{67} + ( 6 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{71} + ( -4 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{73} + ( 6 - 12 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{77} + 12 q^{79} + ( 1 - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{83} + ( 2 - 4 \zeta_{12}^{2} ) q^{85} + ( 6 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{89} + ( -6 + 12 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{91} + 2 q^{95} + ( 4 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{7} + O(q^{10})$$ $$4 q - 12 q^{7} + 12 q^{23} - 4 q^{25} + 24 q^{31} - 24 q^{41} - 12 q^{47} + 20 q^{49} + 24 q^{71} - 16 q^{73} + 48 q^{79} + 24 q^{89} + 8 q^{95} + 16 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$577$$ $$641$$ $$901$$ $$2431$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1441.1
 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i
0 0 0 1.00000i 0 −4.73205 0 0 0
1441.2 0 0 0 1.00000i 0 −1.26795 0 0 0
1441.3 0 0 0 1.00000i 0 −4.73205 0 0 0
1441.4 0 0 0 1.00000i 0 −1.26795 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.k.e 4
3.b odd 2 1 320.2.d.a 4
4.b odd 2 1 2880.2.k.l 4
8.b even 2 1 inner 2880.2.k.e 4
8.d odd 2 1 2880.2.k.l 4
12.b even 2 1 320.2.d.b yes 4
15.d odd 2 1 1600.2.d.h 4
15.e even 4 1 1600.2.f.e 4
15.e even 4 1 1600.2.f.i 4
24.f even 2 1 320.2.d.b yes 4
24.h odd 2 1 320.2.d.a 4
48.i odd 4 1 1280.2.a.b 2
48.i odd 4 1 1280.2.a.p 2
48.k even 4 1 1280.2.a.c 2
48.k even 4 1 1280.2.a.m 2
60.h even 2 1 1600.2.d.b 4
60.l odd 4 1 1600.2.f.d 4
60.l odd 4 1 1600.2.f.h 4
120.i odd 2 1 1600.2.d.h 4
120.m even 2 1 1600.2.d.b 4
120.q odd 4 1 1600.2.f.d 4
120.q odd 4 1 1600.2.f.h 4
120.w even 4 1 1600.2.f.e 4
120.w even 4 1 1600.2.f.i 4
240.t even 4 1 6400.2.a.bf 2
240.t even 4 1 6400.2.a.ck 2
240.bm odd 4 1 6400.2.a.y 2
240.bm odd 4 1 6400.2.a.cd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.d.a 4 3.b odd 2 1
320.2.d.a 4 24.h odd 2 1
320.2.d.b yes 4 12.b even 2 1
320.2.d.b yes 4 24.f even 2 1
1280.2.a.b 2 48.i odd 4 1
1280.2.a.c 2 48.k even 4 1
1280.2.a.m 2 48.k even 4 1
1280.2.a.p 2 48.i odd 4 1
1600.2.d.b 4 60.h even 2 1
1600.2.d.b 4 120.m even 2 1
1600.2.d.h 4 15.d odd 2 1
1600.2.d.h 4 120.i odd 2 1
1600.2.f.d 4 60.l odd 4 1
1600.2.f.d 4 120.q odd 4 1
1600.2.f.e 4 15.e even 4 1
1600.2.f.e 4 120.w even 4 1
1600.2.f.h 4 60.l odd 4 1
1600.2.f.h 4 120.q odd 4 1
1600.2.f.i 4 15.e even 4 1
1600.2.f.i 4 120.w even 4 1
2880.2.k.e 4 1.a even 1 1 trivial
2880.2.k.e 4 8.b even 2 1 inner
2880.2.k.l 4 4.b odd 2 1
2880.2.k.l 4 8.d odd 2 1
6400.2.a.y 2 240.bm odd 4 1
6400.2.a.bf 2 240.t even 4 1
6400.2.a.cd 2 240.bm odd 4 1
6400.2.a.ck 2 240.t even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 6 T_{7} + 6$$ $$T_{11}^{2} + 12$$ $$T_{23}^{2} - 6 T_{23} - 18$$ $$T_{47}^{2} + 6 T_{47} - 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$( 6 + 6 T + T^{2} )^{2}$$
$11$ $$( 12 + T^{2} )^{2}$$
$13$ $$( 12 + T^{2} )^{2}$$
$17$ $$( -12 + T^{2} )^{2}$$
$19$ $$( 4 + T^{2} )^{2}$$
$23$ $$( -18 - 6 T + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 24 - 12 T + T^{2} )^{2}$$
$37$ $$( 36 + T^{2} )^{2}$$
$41$ $$( 24 + 12 T + T^{2} )^{2}$$
$43$ $$4 + 104 T^{2} + T^{4}$$
$47$ $$( -18 + 6 T + T^{2} )^{2}$$
$53$ $$( 108 + T^{2} )^{2}$$
$59$ $$( 36 + T^{2} )^{2}$$
$61$ $$144 + 168 T^{2} + T^{4}$$
$67$ $$4 + 104 T^{2} + T^{4}$$
$71$ $$( -72 - 12 T + T^{2} )^{2}$$
$73$ $$( -92 + 8 T + T^{2} )^{2}$$
$79$ $$( -12 + T )^{4}$$
$83$ $$36 + 24 T^{2} + T^{4}$$
$89$ $$( -12 - 12 T + T^{2} )^{2}$$
$97$ $$( -92 - 8 T + T^{2} )^{2}$$
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