# Properties

 Label 2880.2.f.o Level $2880$ Weight $2$ Character orbit 2880.f 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.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.9969157821$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 480) 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.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{5} + 2 \beta q^{7}+O(q^{10})$$ q + (b + 1) * q^5 + 2*b * q^7 $$q + (\beta + 1) q^{5} + 2 \beta q^{7} + 2 \beta q^{13} + 8 q^{19} - 2 \beta q^{23} + (2 \beta - 3) q^{25} + 6 q^{29} + 8 q^{31} + (2 \beta - 8) q^{35} + 2 \beta q^{37} - 6 q^{41} - 2 \beta q^{43} - 2 \beta q^{47} - 9 q^{49} + 6 \beta q^{53} + 6 q^{61} + (2 \beta - 8) q^{65} + 6 \beta q^{67} - 16 q^{71} - 8 q^{79} - 6 \beta q^{83} - 10 q^{89} - 16 q^{91} + (8 \beta + 8) q^{95} + 4 \beta q^{97} +O(q^{100})$$ q + (b + 1) * q^5 + 2*b * q^7 + 2*b * q^13 + 8 * q^19 - 2*b * q^23 + (2*b - 3) * q^25 + 6 * q^29 + 8 * q^31 + (2*b - 8) * q^35 + 2*b * q^37 - 6 * q^41 - 2*b * q^43 - 2*b * q^47 - 9 * q^49 + 6*b * q^53 + 6 * q^61 + (2*b - 8) * q^65 + 6*b * q^67 - 16 * q^71 - 8 * q^79 - 6*b * q^83 - 10 * q^89 - 16 * q^91 + (8*b + 8) * q^95 + 4*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5}+O(q^{10})$$ 2 * q + 2 * q^5 $$2 q + 2 q^{5} + 16 q^{19} - 6 q^{25} + 12 q^{29} + 16 q^{31} - 16 q^{35} - 12 q^{41} - 18 q^{49} + 12 q^{61} - 16 q^{65} - 32 q^{71} - 16 q^{79} - 20 q^{89} - 32 q^{91} + 16 q^{95}+O(q^{100})$$ 2 * q + 2 * q^5 + 16 * q^19 - 6 * q^25 + 12 * q^29 + 16 * q^31 - 16 * q^35 - 12 * q^41 - 18 * q^49 + 12 * q^61 - 16 * q^65 - 32 * q^71 - 16 * q^79 - 20 * q^89 - 32 * q^91 + 16 * q^95

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1729.1
 − 1.00000i 1.00000i
0 0 0 1.00000 2.00000i 0 4.00000i 0 0 0
1729.2 0 0 0 1.00000 + 2.00000i 0 4.00000i 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
5.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.f.o 2
3.b odd 2 1 960.2.f.d 2
4.b odd 2 1 2880.2.f.m 2
5.b even 2 1 inner 2880.2.f.o 2
8.b even 2 1 1440.2.f.b 2
8.d odd 2 1 1440.2.f.d 2
12.b even 2 1 960.2.f.e 2
15.d odd 2 1 960.2.f.d 2
15.e even 4 1 4800.2.a.e 1
15.e even 4 1 4800.2.a.cp 1
20.d odd 2 1 2880.2.f.m 2
24.f even 2 1 480.2.f.d yes 2
24.h odd 2 1 480.2.f.c 2
40.e odd 2 1 1440.2.f.d 2
40.f even 2 1 1440.2.f.b 2
40.i odd 4 1 7200.2.a.a 1
40.i odd 4 1 7200.2.a.ca 1
40.k even 4 1 7200.2.a.c 1
40.k even 4 1 7200.2.a.by 1
48.i odd 4 1 3840.2.d.p 2
48.i odd 4 1 3840.2.d.q 2
48.k even 4 1 3840.2.d.a 2
48.k even 4 1 3840.2.d.bf 2
60.h even 2 1 960.2.f.e 2
60.l odd 4 1 4800.2.a.c 1
60.l odd 4 1 4800.2.a.cr 1
120.i odd 2 1 480.2.f.c 2
120.m even 2 1 480.2.f.d yes 2
120.q odd 4 1 2400.2.a.o 1
120.q odd 4 1 2400.2.a.t 1
120.w even 4 1 2400.2.a.p 1
120.w even 4 1 2400.2.a.s 1
240.t even 4 1 3840.2.d.a 2
240.t even 4 1 3840.2.d.bf 2
240.bm odd 4 1 3840.2.d.p 2
240.bm odd 4 1 3840.2.d.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.c 2 24.h odd 2 1
480.2.f.c 2 120.i odd 2 1
480.2.f.d yes 2 24.f even 2 1
480.2.f.d yes 2 120.m even 2 1
960.2.f.d 2 3.b odd 2 1
960.2.f.d 2 15.d odd 2 1
960.2.f.e 2 12.b even 2 1
960.2.f.e 2 60.h even 2 1
1440.2.f.b 2 8.b even 2 1
1440.2.f.b 2 40.f even 2 1
1440.2.f.d 2 8.d odd 2 1
1440.2.f.d 2 40.e odd 2 1
2400.2.a.o 1 120.q odd 4 1
2400.2.a.p 1 120.w even 4 1
2400.2.a.s 1 120.w even 4 1
2400.2.a.t 1 120.q odd 4 1
2880.2.f.m 2 4.b odd 2 1
2880.2.f.m 2 20.d odd 2 1
2880.2.f.o 2 1.a even 1 1 trivial
2880.2.f.o 2 5.b even 2 1 inner
3840.2.d.a 2 48.k even 4 1
3840.2.d.a 2 240.t even 4 1
3840.2.d.p 2 48.i odd 4 1
3840.2.d.p 2 240.bm odd 4 1
3840.2.d.q 2 48.i odd 4 1
3840.2.d.q 2 240.bm odd 4 1
3840.2.d.bf 2 48.k even 4 1
3840.2.d.bf 2 240.t even 4 1
4800.2.a.c 1 60.l odd 4 1
4800.2.a.e 1 15.e even 4 1
4800.2.a.cp 1 15.e even 4 1
4800.2.a.cr 1 60.l odd 4 1
7200.2.a.a 1 40.i odd 4 1
7200.2.a.c 1 40.k even 4 1
7200.2.a.by 1 40.k even 4 1
7200.2.a.ca 1 40.i odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11}$$ T11 $$T_{13}^{2} + 16$$ T13^2 + 16 $$T_{17}$$ T17 $$T_{19} - 8$$ T19 - 8 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 5$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2}$$
$19$ $$(T - 8)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 16$$
$53$ $$T^{2} + 144$$
$59$ $$T^{2}$$
$61$ $$(T - 6)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$(T + 16)^{2}$$
$73$ $$T^{2}$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 64$$