# Properties

 Label 80.2.c.a Level $80$ Weight $2$ Character orbit 80.c Analytic conductor $0.639$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 80.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.638803216170$$ 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: $$2$$ Twist minimal: no (minimal twist has level 40) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} + ( -1 + 2 i ) q^{5} -2 i q^{7} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} + ( -1 + 2 i ) q^{5} -2 i q^{7} - q^{9} + 4 q^{11} -4 i q^{13} + ( -4 - 2 i ) q^{15} -4 q^{19} + 4 q^{21} -2 i q^{23} + ( -3 - 4 i ) q^{25} + 4 i q^{27} -2 q^{29} + 8 i q^{33} + ( 4 + 2 i ) q^{35} + 4 i q^{37} + 8 q^{39} + 2 q^{41} -6 i q^{43} + ( 1 - 2 i ) q^{45} + 6 i q^{47} + 3 q^{49} + 4 i q^{53} + ( -4 + 8 i ) q^{55} -8 i q^{57} -12 q^{59} -10 q^{61} + 2 i q^{63} + ( 8 + 4 i ) q^{65} -14 i q^{67} + 4 q^{69} -8 q^{71} -8 i q^{73} + ( 8 - 6 i ) q^{75} -8 i q^{77} + 16 q^{79} -11 q^{81} + 2 i q^{83} -4 i q^{87} -6 q^{89} -8 q^{91} + ( 4 - 8 i ) q^{95} + 16 i q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{5} - 2q^{9} + 8q^{11} - 8q^{15} - 8q^{19} + 8q^{21} - 6q^{25} - 4q^{29} + 8q^{35} + 16q^{39} + 4q^{41} + 2q^{45} + 6q^{49} - 8q^{55} - 24q^{59} - 20q^{61} + 16q^{65} + 8q^{69} - 16q^{71} + 16q^{75} + 32q^{79} - 22q^{81} - 12q^{89} - 16q^{91} + 8q^{95} - 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$21$$ $$31$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 2.00000i 0 −1.00000 2.00000i 0 2.00000i 0 −1.00000 0
49.2 0 2.00000i 0 −1.00000 + 2.00000i 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 80.2.c.a 2
3.b odd 2 1 720.2.f.e 2
4.b odd 2 1 40.2.c.a 2
5.b even 2 1 inner 80.2.c.a 2
5.c odd 4 1 400.2.a.b 1
5.c odd 4 1 400.2.a.g 1
8.b even 2 1 320.2.c.b 2
8.d odd 2 1 320.2.c.c 2
12.b even 2 1 360.2.f.c 2
15.d odd 2 1 720.2.f.e 2
15.e even 4 1 3600.2.a.k 1
15.e even 4 1 3600.2.a.bb 1
16.e even 4 1 1280.2.f.b 2
16.e even 4 1 1280.2.f.e 2
16.f odd 4 1 1280.2.f.a 2
16.f odd 4 1 1280.2.f.f 2
20.d odd 2 1 40.2.c.a 2
20.e even 4 1 200.2.a.b 1
20.e even 4 1 200.2.a.d 1
24.f even 2 1 2880.2.f.h 2
24.h odd 2 1 2880.2.f.i 2
28.d even 2 1 1960.2.g.b 2
40.e odd 2 1 320.2.c.c 2
40.f even 2 1 320.2.c.b 2
40.i odd 4 1 1600.2.a.d 1
40.i odd 4 1 1600.2.a.u 1
40.k even 4 1 1600.2.a.f 1
40.k even 4 1 1600.2.a.v 1
60.h even 2 1 360.2.f.c 2
60.l odd 4 1 1800.2.a.j 1
60.l odd 4 1 1800.2.a.s 1
80.k odd 4 1 1280.2.f.a 2
80.k odd 4 1 1280.2.f.f 2
80.q even 4 1 1280.2.f.b 2
80.q even 4 1 1280.2.f.e 2
120.i odd 2 1 2880.2.f.i 2
120.m even 2 1 2880.2.f.h 2
140.c even 2 1 1960.2.g.b 2
140.j odd 4 1 9800.2.a.d 1
140.j odd 4 1 9800.2.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.c.a 2 4.b odd 2 1
40.2.c.a 2 20.d odd 2 1
80.2.c.a 2 1.a even 1 1 trivial
80.2.c.a 2 5.b even 2 1 inner
200.2.a.b 1 20.e even 4 1
200.2.a.d 1 20.e even 4 1
320.2.c.b 2 8.b even 2 1
320.2.c.b 2 40.f even 2 1
320.2.c.c 2 8.d odd 2 1
320.2.c.c 2 40.e odd 2 1
360.2.f.c 2 12.b even 2 1
360.2.f.c 2 60.h even 2 1
400.2.a.b 1 5.c odd 4 1
400.2.a.g 1 5.c odd 4 1
720.2.f.e 2 3.b odd 2 1
720.2.f.e 2 15.d odd 2 1
1280.2.f.a 2 16.f odd 4 1
1280.2.f.a 2 80.k odd 4 1
1280.2.f.b 2 16.e even 4 1
1280.2.f.b 2 80.q even 4 1
1280.2.f.e 2 16.e even 4 1
1280.2.f.e 2 80.q even 4 1
1280.2.f.f 2 16.f odd 4 1
1280.2.f.f 2 80.k odd 4 1
1600.2.a.d 1 40.i odd 4 1
1600.2.a.f 1 40.k even 4 1
1600.2.a.u 1 40.i odd 4 1
1600.2.a.v 1 40.k even 4 1
1800.2.a.j 1 60.l odd 4 1
1800.2.a.s 1 60.l odd 4 1
1960.2.g.b 2 28.d even 2 1
1960.2.g.b 2 140.c even 2 1
2880.2.f.h 2 24.f even 2 1
2880.2.f.h 2 120.m even 2 1
2880.2.f.i 2 24.h odd 2 1
2880.2.f.i 2 120.i odd 2 1
3600.2.a.k 1 15.e even 4 1
3600.2.a.bb 1 15.e even 4 1
9800.2.a.d 1 140.j odd 4 1
9800.2.a.bf 1 140.j odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(80, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$5 + 2 T + T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$4 + T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$16 + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$36 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$16 + T^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$196 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$64 + T^{2}$$
$79$ $$( -16 + T )^{2}$$
$83$ $$4 + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$256 + T^{2}$$