# Properties

 Label 2880.2.f.c 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: $$1$$ Twist minimal: no (minimal twist has level 30) 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.

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

## 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 −2.00000 1.00000i 0 2.00000i 0 0 0
1729.2 0 0 0 −2.00000 + 1.00000i 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
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.c 2
3.b odd 2 1 960.2.f.i 2
4.b odd 2 1 2880.2.f.e 2
5.b even 2 1 inner 2880.2.f.c 2
8.b even 2 1 720.2.f.f 2
8.d odd 2 1 90.2.c.a 2
12.b even 2 1 960.2.f.h 2
15.d odd 2 1 960.2.f.i 2
15.e even 4 1 4800.2.a.m 1
15.e even 4 1 4800.2.a.cj 1
20.d odd 2 1 2880.2.f.e 2
24.f even 2 1 30.2.c.a 2
24.h odd 2 1 240.2.f.a 2
40.e odd 2 1 90.2.c.a 2
40.f even 2 1 720.2.f.f 2
40.i odd 4 1 3600.2.a.o 1
40.i odd 4 1 3600.2.a.bg 1
40.k even 4 1 450.2.a.b 1
40.k even 4 1 450.2.a.f 1
48.i odd 4 1 3840.2.d.j 2
48.i odd 4 1 3840.2.d.x 2
48.k even 4 1 3840.2.d.g 2
48.k even 4 1 3840.2.d.y 2
60.h even 2 1 960.2.f.h 2
60.l odd 4 1 4800.2.a.l 1
60.l odd 4 1 4800.2.a.cg 1
72.l even 6 2 810.2.i.e 4
72.p odd 6 2 810.2.i.b 4
120.i odd 2 1 240.2.f.a 2
120.m even 2 1 30.2.c.a 2
120.q odd 4 1 150.2.a.a 1
120.q odd 4 1 150.2.a.c 1
120.w even 4 1 1200.2.a.g 1
120.w even 4 1 1200.2.a.m 1
168.e odd 2 1 1470.2.g.g 2
168.v even 6 2 1470.2.n.h 4
168.be odd 6 2 1470.2.n.a 4
240.t even 4 1 3840.2.d.g 2
240.t even 4 1 3840.2.d.y 2
240.bm odd 4 1 3840.2.d.j 2
240.bm odd 4 1 3840.2.d.x 2
360.z odd 6 2 810.2.i.b 4
360.bd even 6 2 810.2.i.e 4
840.b odd 2 1 1470.2.g.g 2
840.bm even 4 1 7350.2.a.bg 1
840.bm even 4 1 7350.2.a.cc 1
840.ct odd 6 2 1470.2.n.a 4
840.cv even 6 2 1470.2.n.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 24.f even 2 1
30.2.c.a 2 120.m even 2 1
90.2.c.a 2 8.d odd 2 1
90.2.c.a 2 40.e odd 2 1
150.2.a.a 1 120.q odd 4 1
150.2.a.c 1 120.q odd 4 1
240.2.f.a 2 24.h odd 2 1
240.2.f.a 2 120.i odd 2 1
450.2.a.b 1 40.k even 4 1
450.2.a.f 1 40.k even 4 1
720.2.f.f 2 8.b even 2 1
720.2.f.f 2 40.f even 2 1
810.2.i.b 4 72.p odd 6 2
810.2.i.b 4 360.z odd 6 2
810.2.i.e 4 72.l even 6 2
810.2.i.e 4 360.bd even 6 2
960.2.f.h 2 12.b even 2 1
960.2.f.h 2 60.h even 2 1
960.2.f.i 2 3.b odd 2 1
960.2.f.i 2 15.d odd 2 1
1200.2.a.g 1 120.w even 4 1
1200.2.a.m 1 120.w even 4 1
1470.2.g.g 2 168.e odd 2 1
1470.2.g.g 2 840.b odd 2 1
1470.2.n.a 4 168.be odd 6 2
1470.2.n.a 4 840.ct odd 6 2
1470.2.n.h 4 168.v even 6 2
1470.2.n.h 4 840.cv even 6 2
2880.2.f.c 2 1.a even 1 1 trivial
2880.2.f.c 2 5.b even 2 1 inner
2880.2.f.e 2 4.b odd 2 1
2880.2.f.e 2 20.d odd 2 1
3600.2.a.o 1 40.i odd 4 1
3600.2.a.bg 1 40.i odd 4 1
3840.2.d.g 2 48.k even 4 1
3840.2.d.g 2 240.t even 4 1
3840.2.d.j 2 48.i odd 4 1
3840.2.d.j 2 240.bm odd 4 1
3840.2.d.x 2 48.i odd 4 1
3840.2.d.x 2 240.bm odd 4 1
3840.2.d.y 2 48.k even 4 1
3840.2.d.y 2 240.t even 4 1
4800.2.a.l 1 60.l odd 4 1
4800.2.a.m 1 15.e even 4 1
4800.2.a.cg 1 60.l odd 4 1
4800.2.a.cj 1 15.e even 4 1
7350.2.a.bg 1 840.bm even 4 1
7350.2.a.cc 1 840.bm even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11} + 2$$ T11 + 2 $$T_{13}^{2} + 36$$ T13^2 + 36 $$T_{17}^{2} + 4$$ T17^2 + 4 $$T_{19}$$ T19 $$T_{29}$$ T29

## Hecke characteristic polynomials

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