# Properties

 Label 2880.2.k.g Level $2880$ Weight $2$ Character orbit 2880.k Analytic conductor $22.997$ Analytic rank $0$ Dimension $4$ 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.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: $$2^{4}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{3} q^{5} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} +O(q^{10})$$ $$q + \zeta_{12}^{3} q^{5} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + ( 2 - 4 \zeta_{12}^{2} ) q^{11} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{17} + 2 \zeta_{12}^{3} q^{19} -6 q^{23} - q^{25} + 6 \zeta_{12}^{3} q^{29} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{31} + ( 2 - 4 \zeta_{12}^{2} ) q^{35} + ( 4 - 8 \zeta_{12}^{2} ) q^{37} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{41} + 4 \zeta_{12}^{3} q^{43} -6 q^{47} + 5 q^{49} + 6 \zeta_{12}^{3} q^{53} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{55} + ( -6 + 12 \zeta_{12}^{2} ) q^{59} + ( -8 + 16 \zeta_{12}^{2} ) q^{61} -8 \zeta_{12}^{3} q^{67} -12 q^{71} + 2 q^{73} + 12 \zeta_{12}^{3} q^{77} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{79} + ( -4 + 8 \zeta_{12}^{2} ) q^{83} + ( -4 + 8 \zeta_{12}^{2} ) q^{85} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{89} -2 q^{95} -14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 24q^{23} - 4q^{25} - 24q^{47} + 20q^{49} - 48q^{71} + 8q^{73} - 8q^{95} - 56q^{97} + O(q^{100})$$

## 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 −3.46410 0 0 0
1441.2 0 0 0 1.00000i 0 3.46410 0 0 0
1441.3 0 0 0 1.00000i 0 −3.46410 0 0 0
1441.4 0 0 0 1.00000i 0 3.46410 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
12.b even 2 1 inner
24.f 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.g 4
3.b odd 2 1 2880.2.k.j yes 4
4.b odd 2 1 2880.2.k.j yes 4
8.b even 2 1 inner 2880.2.k.g 4
8.d odd 2 1 2880.2.k.j yes 4
12.b even 2 1 inner 2880.2.k.g 4
24.f even 2 1 inner 2880.2.k.g 4
24.h odd 2 1 2880.2.k.j yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2880.2.k.g 4 1.a even 1 1 trivial
2880.2.k.g 4 8.b even 2 1 inner
2880.2.k.g 4 12.b even 2 1 inner
2880.2.k.g 4 24.f even 2 1 inner
2880.2.k.j yes 4 3.b odd 2 1
2880.2.k.j yes 4 4.b odd 2 1
2880.2.k.j yes 4 8.d odd 2 1
2880.2.k.j yes 4 24.h odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$( -12 + T^{2} )^{2}$$
$11$ $$( 12 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( -48 + T^{2} )^{2}$$
$19$ $$( 4 + T^{2} )^{2}$$
$23$ $$( 6 + T )^{4}$$
$29$ $$( 36 + T^{2} )^{2}$$
$31$ $$( -48 + T^{2} )^{2}$$
$37$ $$( 48 + T^{2} )^{2}$$
$41$ $$( -48 + T^{2} )^{2}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$( 6 + T )^{4}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( 108 + T^{2} )^{2}$$
$61$ $$( 192 + T^{2} )^{2}$$
$67$ $$( 64 + T^{2} )^{2}$$
$71$ $$( 12 + T )^{4}$$
$73$ $$( -2 + T )^{4}$$
$79$ $$( -48 + T^{2} )^{2}$$
$83$ $$( 48 + T^{2} )^{2}$$
$89$ $$( -48 + T^{2} )^{2}$$
$97$ $$( 14 + T )^{4}$$