# Properties

 Label 980.2.i.i Level $980$ Weight $2$ Character orbit 980.i Analytic conductor $7.825$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 980.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82533939809$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} -\zeta_{6} q^{9} + 2 q^{13} + 2 q^{15} + ( 6 - 6 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} -6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 4 q^{27} + 6 q^{29} + ( 4 - 4 \zeta_{6} ) q^{31} -2 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{39} + 6 q^{41} -10 q^{43} + ( 1 - \zeta_{6} ) q^{45} + 6 \zeta_{6} q^{47} -12 \zeta_{6} q^{51} + ( 6 - 6 \zeta_{6} ) q^{53} + 8 q^{57} + ( -12 + 12 \zeta_{6} ) q^{59} -2 \zeta_{6} q^{61} + 2 \zeta_{6} q^{65} + ( -2 + 2 \zeta_{6} ) q^{67} -12 q^{69} -12 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} + 2 \zeta_{6} q^{75} -8 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} + 6 q^{83} + 6 q^{85} + ( 12 - 12 \zeta_{6} ) q^{87} + 6 \zeta_{6} q^{89} -8 \zeta_{6} q^{93} + ( -4 + 4 \zeta_{6} ) q^{95} + 2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + q^{5} - q^{9} + O(q^{10})$$ $$2q + 2q^{3} + q^{5} - q^{9} + 4q^{13} + 4q^{15} + 6q^{17} + 4q^{19} - 6q^{23} - q^{25} + 8q^{27} + 12q^{29} + 4q^{31} - 2q^{37} + 4q^{39} + 12q^{41} - 20q^{43} + q^{45} + 6q^{47} - 12q^{51} + 6q^{53} + 16q^{57} - 12q^{59} - 2q^{61} + 2q^{65} - 2q^{67} - 24q^{69} - 24q^{71} - 2q^{73} + 2q^{75} - 8q^{79} + 11q^{81} + 12q^{83} + 12q^{85} + 12q^{87} + 6q^{89} - 8q^{93} - 4q^{95} + 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.00000 1.73205i 0 0.500000 + 0.866025i 0 0 0 −0.500000 0.866025i 0
961.1 0 1.00000 + 1.73205i 0 0.500000 0.866025i 0 0 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.i.i 2
7.b odd 2 1 980.2.i.c 2
7.c even 3 1 20.2.a.a 1
7.c even 3 1 inner 980.2.i.i 2
7.d odd 6 1 980.2.a.h 1
7.d odd 6 1 980.2.i.c 2
21.g even 6 1 8820.2.a.g 1
21.h odd 6 1 180.2.a.a 1
28.f even 6 1 3920.2.a.h 1
28.g odd 6 1 80.2.a.b 1
35.i odd 6 1 4900.2.a.e 1
35.j even 6 1 100.2.a.a 1
35.k even 12 2 4900.2.e.f 2
35.l odd 12 2 100.2.c.a 2
56.k odd 6 1 320.2.a.a 1
56.p even 6 1 320.2.a.f 1
63.g even 3 1 1620.2.i.h 2
63.h even 3 1 1620.2.i.h 2
63.j odd 6 1 1620.2.i.b 2
63.n odd 6 1 1620.2.i.b 2
77.h odd 6 1 2420.2.a.a 1
84.n even 6 1 720.2.a.h 1
91.r even 6 1 3380.2.a.c 1
91.z odd 12 2 3380.2.f.b 2
105.o odd 6 1 900.2.a.b 1
105.x even 12 2 900.2.d.c 2
112.u odd 12 2 1280.2.d.g 2
112.w even 12 2 1280.2.d.c 2
119.j even 6 1 5780.2.a.f 1
119.n even 12 2 5780.2.c.a 2
133.r odd 6 1 7220.2.a.f 1
140.p odd 6 1 400.2.a.c 1
140.w even 12 2 400.2.c.b 2
168.s odd 6 1 2880.2.a.m 1
168.v even 6 1 2880.2.a.f 1
280.bf even 6 1 1600.2.a.c 1
280.bi odd 6 1 1600.2.a.w 1
280.br even 12 2 1600.2.c.e 2
280.bt odd 12 2 1600.2.c.d 2
308.n even 6 1 9680.2.a.ba 1
420.ba even 6 1 3600.2.a.be 1
420.bp odd 12 2 3600.2.f.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 7.c even 3 1
80.2.a.b 1 28.g odd 6 1
100.2.a.a 1 35.j even 6 1
100.2.c.a 2 35.l odd 12 2
180.2.a.a 1 21.h odd 6 1
320.2.a.a 1 56.k odd 6 1
320.2.a.f 1 56.p even 6 1
400.2.a.c 1 140.p odd 6 1
400.2.c.b 2 140.w even 12 2
720.2.a.h 1 84.n even 6 1
900.2.a.b 1 105.o odd 6 1
900.2.d.c 2 105.x even 12 2
980.2.a.h 1 7.d odd 6 1
980.2.i.c 2 7.b odd 2 1
980.2.i.c 2 7.d odd 6 1
980.2.i.i 2 1.a even 1 1 trivial
980.2.i.i 2 7.c even 3 1 inner
1280.2.d.c 2 112.w even 12 2
1280.2.d.g 2 112.u odd 12 2
1600.2.a.c 1 280.bf even 6 1
1600.2.a.w 1 280.bi odd 6 1
1600.2.c.d 2 280.bt odd 12 2
1600.2.c.e 2 280.br even 12 2
1620.2.i.b 2 63.j odd 6 1
1620.2.i.b 2 63.n odd 6 1
1620.2.i.h 2 63.g even 3 1
1620.2.i.h 2 63.h even 3 1
2420.2.a.a 1 77.h odd 6 1
2880.2.a.f 1 168.v even 6 1
2880.2.a.m 1 168.s odd 6 1
3380.2.a.c 1 91.r even 6 1
3380.2.f.b 2 91.z odd 12 2
3600.2.a.be 1 420.ba even 6 1
3600.2.f.j 2 420.bp odd 12 2
3920.2.a.h 1 28.f even 6 1
4900.2.a.e 1 35.i odd 6 1
4900.2.e.f 2 35.k even 12 2
5780.2.a.f 1 119.j even 6 1
5780.2.c.a 2 119.n even 12 2
7220.2.a.f 1 133.r odd 6 1
8820.2.a.g 1 21.g even 6 1
9680.2.a.ba 1 308.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(980, [\chi])$$:

 $$T_{3}^{2} - 2 T_{3} + 4$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 - 2 T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$16 - 4 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$4 + 2 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( 10 + T )^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$4 + 2 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$4 + 2 T + T^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$( -2 + T )^{2}$$