# Properties

 Label 480.2.f.b Level $480$ Weight $2$ Character orbit 480.f Analytic conductor $3.833$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$480 = 2^{5} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 480.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.83281929702$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} + ( -2 - i ) q^{5} -2 i q^{7} - q^{9} +O(q^{10})$$ $$q + i q^{3} + ( -2 - i ) q^{5} -2 i q^{7} - q^{9} + 6 q^{11} + 2 i q^{13} + ( 1 - 2 i ) q^{15} -6 i q^{17} + 4 q^{19} + 2 q^{21} -8 i q^{23} + ( 3 + 4 i ) q^{25} -i q^{27} + 8 q^{31} + 6 i q^{33} + ( -2 + 4 i ) q^{35} + 2 i q^{37} -2 q^{39} -6 q^{41} + 4 i q^{43} + ( 2 + i ) q^{45} + 4 i q^{47} + 3 q^{49} + 6 q^{51} -6 i q^{53} + ( -12 - 6 i ) q^{55} + 4 i q^{57} -6 q^{59} -6 q^{61} + 2 i q^{63} + ( 2 - 4 i ) q^{65} + 8 q^{69} + 4 q^{71} -12 i q^{73} + ( -4 + 3 i ) q^{75} -12 i q^{77} -8 q^{79} + q^{81} + 12 i q^{83} + ( -6 + 12 i ) q^{85} -14 q^{89} + 4 q^{91} + 8 i q^{93} + ( -8 - 4 i ) q^{95} + 8 i q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5} - 2 q^{9} + O(q^{10})$$ $$2 q - 4 q^{5} - 2 q^{9} + 12 q^{11} + 2 q^{15} + 8 q^{19} + 4 q^{21} + 6 q^{25} + 16 q^{31} - 4 q^{35} - 4 q^{39} - 12 q^{41} + 4 q^{45} + 6 q^{49} + 12 q^{51} - 24 q^{55} - 12 q^{59} - 12 q^{61} + 4 q^{65} + 16 q^{69} + 8 q^{71} - 8 q^{75} - 16 q^{79} + 2 q^{81} - 12 q^{85} - 28 q^{89} + 8 q^{91} - 16 q^{95} - 12 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$161$$ $$421$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 1.00000i 0 −2.00000 + 1.00000i 0 2.00000i 0 −1.00000 0
289.2 0 1.00000i 0 −2.00000 1.00000i 0 2.00000i 0 −1.00000 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 480.2.f.b yes 2
3.b odd 2 1 1440.2.f.e 2
4.b odd 2 1 480.2.f.a 2
5.b even 2 1 inner 480.2.f.b yes 2
5.c odd 4 1 2400.2.a.d 1
5.c odd 4 1 2400.2.a.bf 1
8.b even 2 1 960.2.f.g 2
8.d odd 2 1 960.2.f.j 2
12.b even 2 1 1440.2.f.g 2
15.d odd 2 1 1440.2.f.e 2
15.e even 4 1 7200.2.a.j 1
15.e even 4 1 7200.2.a.bl 1
16.e even 4 1 3840.2.d.e 2
16.e even 4 1 3840.2.d.bc 2
16.f odd 4 1 3840.2.d.k 2
16.f odd 4 1 3840.2.d.u 2
20.d odd 2 1 480.2.f.a 2
20.e even 4 1 2400.2.a.c 1
20.e even 4 1 2400.2.a.be 1
24.f even 2 1 2880.2.f.a 2
24.h odd 2 1 2880.2.f.g 2
40.e odd 2 1 960.2.f.j 2
40.f even 2 1 960.2.f.g 2
40.i odd 4 1 4800.2.a.z 1
40.i odd 4 1 4800.2.a.bt 1
40.k even 4 1 4800.2.a.ba 1
40.k even 4 1 4800.2.a.bu 1
60.h even 2 1 1440.2.f.g 2
60.l odd 4 1 7200.2.a.p 1
60.l odd 4 1 7200.2.a.br 1
80.k odd 4 1 3840.2.d.k 2
80.k odd 4 1 3840.2.d.u 2
80.q even 4 1 3840.2.d.e 2
80.q even 4 1 3840.2.d.bc 2
120.i odd 2 1 2880.2.f.g 2
120.m even 2 1 2880.2.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.a 2 4.b odd 2 1
480.2.f.a 2 20.d odd 2 1
480.2.f.b yes 2 1.a even 1 1 trivial
480.2.f.b yes 2 5.b even 2 1 inner
960.2.f.g 2 8.b even 2 1
960.2.f.g 2 40.f even 2 1
960.2.f.j 2 8.d odd 2 1
960.2.f.j 2 40.e odd 2 1
1440.2.f.e 2 3.b odd 2 1
1440.2.f.e 2 15.d odd 2 1
1440.2.f.g 2 12.b even 2 1
1440.2.f.g 2 60.h even 2 1
2400.2.a.c 1 20.e even 4 1
2400.2.a.d 1 5.c odd 4 1
2400.2.a.be 1 20.e even 4 1
2400.2.a.bf 1 5.c odd 4 1
2880.2.f.a 2 24.f even 2 1
2880.2.f.a 2 120.m even 2 1
2880.2.f.g 2 24.h odd 2 1
2880.2.f.g 2 120.i odd 2 1
3840.2.d.e 2 16.e even 4 1
3840.2.d.e 2 80.q even 4 1
3840.2.d.k 2 16.f odd 4 1
3840.2.d.k 2 80.k odd 4 1
3840.2.d.u 2 16.f odd 4 1
3840.2.d.u 2 80.k odd 4 1
3840.2.d.bc 2 16.e even 4 1
3840.2.d.bc 2 80.q even 4 1
4800.2.a.z 1 40.i odd 4 1
4800.2.a.ba 1 40.k even 4 1
4800.2.a.bt 1 40.i odd 4 1
4800.2.a.bu 1 40.k even 4 1
7200.2.a.j 1 15.e even 4 1
7200.2.a.p 1 60.l odd 4 1
7200.2.a.bl 1 15.e even 4 1
7200.2.a.br 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(480, [\chi])$$:

 $$T_{7}^{2} + 4$$ $$T_{11} - 6$$ $$T_{19} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$5 + 4 T + T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( -6 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$16 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$( 6 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$( -4 + T )^{2}$$
$73$ $$144 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 14 + T )^{2}$$
$97$ $$64 + T^{2}$$