# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 960) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -i q^{5} +O(q^{10})$$ $$q -i q^{5} + 6 i q^{13} + 2 i q^{19} -6 q^{23} - q^{25} -6 i q^{29} + 6 i q^{37} -6 q^{41} -8 i q^{43} -6 q^{47} -7 q^{49} + 6 i q^{53} -12 i q^{59} + 12 i q^{61} + 6 q^{65} + 4 i q^{67} -12 q^{71} + 2 q^{73} + 12 i q^{83} + 6 q^{89} + 2 q^{95} + 10 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 12q^{23} - 2q^{25} - 12q^{41} - 12q^{47} - 14q^{49} + 12q^{65} - 24q^{71} + 4q^{73} + 12q^{89} + 4q^{95} + 20q^{97} + O(q^{100})$$

## 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
 1.00000i − 1.00000i
0 0 0 1.00000i 0 0 0 0 0
1441.2 0 0 0 1.00000i 0 0 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.b 2
3.b odd 2 1 960.2.k.c yes 2
4.b odd 2 1 2880.2.k.c 2
8.b even 2 1 inner 2880.2.k.b 2
8.d odd 2 1 2880.2.k.c 2
12.b even 2 1 960.2.k.b 2
15.d odd 2 1 4800.2.k.d 2
15.e even 4 1 4800.2.d.d 2
15.e even 4 1 4800.2.d.e 2
24.f even 2 1 960.2.k.b 2
24.h odd 2 1 960.2.k.c yes 2
48.i odd 4 1 3840.2.a.e 1
48.i odd 4 1 3840.2.a.z 1
48.k even 4 1 3840.2.a.j 1
48.k even 4 1 3840.2.a.q 1
60.h even 2 1 4800.2.k.e 2
60.l odd 4 1 4800.2.d.a 2
60.l odd 4 1 4800.2.d.h 2
120.i odd 2 1 4800.2.k.d 2
120.m even 2 1 4800.2.k.e 2
120.q odd 4 1 4800.2.d.a 2
120.q odd 4 1 4800.2.d.h 2
120.w even 4 1 4800.2.d.d 2
120.w even 4 1 4800.2.d.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.k.b 2 12.b even 2 1
960.2.k.b 2 24.f even 2 1
960.2.k.c yes 2 3.b odd 2 1
960.2.k.c yes 2 24.h odd 2 1
2880.2.k.b 2 1.a even 1 1 trivial
2880.2.k.b 2 8.b even 2 1 inner
2880.2.k.c 2 4.b odd 2 1
2880.2.k.c 2 8.d odd 2 1
3840.2.a.e 1 48.i odd 4 1
3840.2.a.j 1 48.k even 4 1
3840.2.a.q 1 48.k even 4 1
3840.2.a.z 1 48.i odd 4 1
4800.2.d.a 2 60.l odd 4 1
4800.2.d.a 2 120.q odd 4 1
4800.2.d.d 2 15.e even 4 1
4800.2.d.d 2 120.w even 4 1
4800.2.d.e 2 15.e even 4 1
4800.2.d.e 2 120.w even 4 1
4800.2.d.h 2 60.l odd 4 1
4800.2.d.h 2 120.q odd 4 1
4800.2.k.d 2 15.d odd 2 1
4800.2.k.d 2 120.i odd 2 1
4800.2.k.e 2 60.h even 2 1
4800.2.k.e 2 120.m even 2 1

## Hecke kernels

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

 $$T_{7}$$ $$T_{11}$$ $$T_{23} + 6$$ $$T_{47} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$4 + T^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$36 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$( 6 + T )^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$144 + T^{2}$$
$61$ $$144 + T^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$( -10 + T )^{2}$$