# Properties

 Label 980.2.i.g 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{11} + 5 q^{13} - q^{15} + ( 1 - \zeta_{6} ) q^{17} -6 \zeta_{6} q^{19} + 4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 5 q^{27} + 3 q^{29} + ( 2 - 2 \zeta_{6} ) q^{31} -\zeta_{6} q^{33} -8 \zeta_{6} q^{37} + ( 5 - 5 \zeta_{6} ) q^{39} + 10 q^{41} -2 q^{43} + ( 2 - 2 \zeta_{6} ) q^{45} -7 \zeta_{6} q^{47} -\zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{53} - q^{55} -6 q^{57} + ( 14 - 14 \zeta_{6} ) q^{59} -8 \zeta_{6} q^{61} -5 \zeta_{6} q^{65} + ( -14 + 14 \zeta_{6} ) q^{67} + 4 q^{69} + ( -10 + 10 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} + 11 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 4 q^{83} - q^{85} + ( 3 - 3 \zeta_{6} ) q^{87} + 4 \zeta_{6} q^{89} -2 \zeta_{6} q^{93} + ( -6 + 6 \zeta_{6} ) q^{95} + 3 q^{97} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - q^{5} + 2q^{9} + O(q^{10})$$ $$2q + q^{3} - q^{5} + 2q^{9} + q^{11} + 10q^{13} - 2q^{15} + q^{17} - 6q^{19} + 4q^{23} - q^{25} + 10q^{27} + 6q^{29} + 2q^{31} - q^{33} - 8q^{37} + 5q^{39} + 20q^{41} - 4q^{43} + 2q^{45} - 7q^{47} - q^{51} + 2q^{53} - 2q^{55} - 12q^{57} + 14q^{59} - 8q^{61} - 5q^{65} - 14q^{67} + 8q^{69} - 10q^{73} + q^{75} + 11q^{79} - q^{81} + 8q^{83} - 2q^{85} + 3q^{87} + 4q^{89} - 2q^{93} - 6q^{95} + 6q^{97} + 4q^{99} + O(q^{100})$$

## 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 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 1.00000 + 1.73205i 0
961.1 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 1.00000 1.73205i 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.g 2
7.b odd 2 1 980.2.i.e 2
7.c even 3 1 980.2.a.d 1
7.c even 3 1 inner 980.2.i.g 2
7.d odd 6 1 980.2.a.f yes 1
7.d odd 6 1 980.2.i.e 2
21.g even 6 1 8820.2.a.v 1
21.h odd 6 1 8820.2.a.i 1
28.f even 6 1 3920.2.a.n 1
28.g odd 6 1 3920.2.a.z 1
35.i odd 6 1 4900.2.a.h 1
35.j even 6 1 4900.2.a.o 1
35.k even 12 2 4900.2.e.k 2
35.l odd 12 2 4900.2.e.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.d 1 7.c even 3 1
980.2.a.f yes 1 7.d odd 6 1
980.2.i.e 2 7.b odd 2 1
980.2.i.e 2 7.d odd 6 1
980.2.i.g 2 1.a even 1 1 trivial
980.2.i.g 2 7.c even 3 1 inner
3920.2.a.n 1 28.f even 6 1
3920.2.a.z 1 28.g odd 6 1
4900.2.a.h 1 35.i odd 6 1
4900.2.a.o 1 35.j even 6 1
4900.2.e.j 2 35.l odd 12 2
4900.2.e.k 2 35.k even 12 2
8820.2.a.i 1 21.h odd 6 1
8820.2.a.v 1 21.g even 6 1

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{11}^{2} - T_{11} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$( -5 + T )^{2}$$
$17$ $$1 - T + T^{2}$$
$19$ $$36 + 6 T + T^{2}$$
$23$ $$16 - 4 T + T^{2}$$
$29$ $$( -3 + T )^{2}$$
$31$ $$4 - 2 T + T^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$( 2 + T )^{2}$$
$47$ $$49 + 7 T + T^{2}$$
$53$ $$4 - 2 T + T^{2}$$
$59$ $$196 - 14 T + T^{2}$$
$61$ $$64 + 8 T + T^{2}$$
$67$ $$196 + 14 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$100 + 10 T + T^{2}$$
$79$ $$121 - 11 T + T^{2}$$
$83$ $$( -4 + T )^{2}$$
$89$ $$16 - 4 T + T^{2}$$
$97$ $$( -3 + T )^{2}$$