# Properties

 Label 2880.2.d.b Level $2880$ Weight $2$ Character orbit 2880.d 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.d (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: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 i ) q^{5} + 2 i q^{7} +O(q^{10})$$ $$q + ( -1 + 2 i ) q^{5} + 2 i q^{7} -2 i q^{11} + 2 q^{13} -4 i q^{17} -2 i q^{19} -6 i q^{23} + ( -3 - 4 i ) q^{25} + 4 i q^{29} + 8 q^{31} + ( -4 - 2 i ) q^{35} + 10 q^{37} + 6 q^{41} + 4 q^{43} -6 i q^{47} + 3 q^{49} -10 q^{53} + ( 4 + 2 i ) q^{55} + 6 i q^{59} + 8 i q^{61} + ( -2 + 4 i ) q^{65} -12 q^{67} + 4 q^{77} + 16 q^{79} -4 q^{83} + ( 8 + 4 i ) q^{85} -10 q^{89} + 4 i q^{91} + ( 4 + 2 i ) q^{95} + 16 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} + O(q^{10})$$ $$2q - 2q^{5} + 4q^{13} - 6q^{25} + 16q^{31} - 8q^{35} + 20q^{37} + 12q^{41} + 8q^{43} + 6q^{49} - 20q^{53} + 8q^{55} - 4q^{65} - 24q^{67} + 8q^{77} + 32q^{79} - 8q^{83} + 16q^{85} - 20q^{89} + 8q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 − 1.00000i 1.00000i
0 0 0 −1.00000 2.00000i 0 2.00000i 0 0 0
289.2 0 0 0 −1.00000 + 2.00000i 0 2.00000i 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
40.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.d.b 2
3.b odd 2 1 960.2.d.b yes 2
4.b odd 2 1 2880.2.d.a 2
5.b even 2 1 2880.2.d.d 2
8.b even 2 1 2880.2.d.d 2
8.d odd 2 1 2880.2.d.c 2
12.b even 2 1 960.2.d.d yes 2
15.d odd 2 1 960.2.d.c yes 2
15.e even 4 1 4800.2.k.b 2
15.e even 4 1 4800.2.k.g 2
20.d odd 2 1 2880.2.d.c 2
24.f even 2 1 960.2.d.a 2
24.h odd 2 1 960.2.d.c yes 2
40.e odd 2 1 2880.2.d.a 2
40.f even 2 1 inner 2880.2.d.b 2
48.i odd 4 1 3840.2.f.b 2
48.i odd 4 1 3840.2.f.c 2
48.k even 4 1 3840.2.f.a 2
48.k even 4 1 3840.2.f.d 2
60.h even 2 1 960.2.d.a 2
60.l odd 4 1 4800.2.k.c 2
60.l odd 4 1 4800.2.k.f 2
120.i odd 2 1 960.2.d.b yes 2
120.m even 2 1 960.2.d.d yes 2
120.q odd 4 1 4800.2.k.c 2
120.q odd 4 1 4800.2.k.f 2
120.w even 4 1 4800.2.k.b 2
120.w even 4 1 4800.2.k.g 2
240.t even 4 1 3840.2.f.a 2
240.t even 4 1 3840.2.f.d 2
240.bm odd 4 1 3840.2.f.b 2
240.bm odd 4 1 3840.2.f.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.d.a 2 24.f even 2 1
960.2.d.a 2 60.h even 2 1
960.2.d.b yes 2 3.b odd 2 1
960.2.d.b yes 2 120.i odd 2 1
960.2.d.c yes 2 15.d odd 2 1
960.2.d.c yes 2 24.h odd 2 1
960.2.d.d yes 2 12.b even 2 1
960.2.d.d yes 2 120.m even 2 1
2880.2.d.a 2 4.b odd 2 1
2880.2.d.a 2 40.e odd 2 1
2880.2.d.b 2 1.a even 1 1 trivial
2880.2.d.b 2 40.f even 2 1 inner
2880.2.d.c 2 8.d odd 2 1
2880.2.d.c 2 20.d odd 2 1
2880.2.d.d 2 5.b even 2 1
2880.2.d.d 2 8.b even 2 1
3840.2.f.a 2 48.k even 4 1
3840.2.f.a 2 240.t even 4 1
3840.2.f.b 2 48.i odd 4 1
3840.2.f.b 2 240.bm odd 4 1
3840.2.f.c 2 48.i odd 4 1
3840.2.f.c 2 240.bm odd 4 1
3840.2.f.d 2 48.k even 4 1
3840.2.f.d 2 240.t even 4 1
4800.2.k.b 2 15.e even 4 1
4800.2.k.b 2 120.w even 4 1
4800.2.k.c 2 60.l odd 4 1
4800.2.k.c 2 120.q odd 4 1
4800.2.k.f 2 60.l odd 4 1
4800.2.k.f 2 120.q odd 4 1
4800.2.k.g 2 15.e even 4 1
4800.2.k.g 2 120.w even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ $$T_{11}^{2} + 4$$ $$T_{13} - 2$$ $$T_{31} - 8$$ $$T_{41} - 6$$ $$T_{43} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 + 2 T + T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$16 + T^{2}$$
$19$ $$4 + T^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$16 + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$( 10 + T )^{2}$$
$59$ $$36 + T^{2}$$
$61$ $$64 + T^{2}$$
$67$ $$( 12 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$( -16 + T )^{2}$$
$83$ $$( 4 + T )^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$256 + T^{2}$$