# Properties

 Label 980.2.i.f 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: no (minimal twist has level 140) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( -6 + 6 \zeta_{6} ) q^{11} -2 q^{13} - q^{15} + ( -6 + 6 \zeta_{6} ) q^{17} + 8 \zeta_{6} q^{19} -3 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 5 q^{27} + 3 q^{29} + ( 2 - 2 \zeta_{6} ) q^{31} + 6 \zeta_{6} q^{33} -8 \zeta_{6} q^{37} + ( -2 + 2 \zeta_{6} ) q^{39} + 3 q^{41} + 5 q^{43} + ( 2 - 2 \zeta_{6} ) q^{45} + 6 \zeta_{6} q^{51} + ( -12 + 12 \zeta_{6} ) q^{53} + 6 q^{55} + 8 q^{57} -\zeta_{6} q^{61} + 2 \zeta_{6} q^{65} + ( 7 - 7 \zeta_{6} ) q^{67} -3 q^{69} + ( -10 + 10 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} + 4 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -3 q^{83} + 6 q^{85} + ( 3 - 3 \zeta_{6} ) q^{87} -3 \zeta_{6} q^{89} -2 \zeta_{6} q^{93} + ( 8 - 8 \zeta_{6} ) q^{95} + 10 q^{97} -12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - q^{5} + 2q^{9} + O(q^{10})$$ $$2q + q^{3} - q^{5} + 2q^{9} - 6q^{11} - 4q^{13} - 2q^{15} - 6q^{17} + 8q^{19} - 3q^{23} - q^{25} + 10q^{27} + 6q^{29} + 2q^{31} + 6q^{33} - 8q^{37} - 2q^{39} + 6q^{41} + 10q^{43} + 2q^{45} + 6q^{51} - 12q^{53} + 12q^{55} + 16q^{57} - q^{61} + 2q^{65} + 7q^{67} - 6q^{69} - 10q^{73} + q^{75} + 4q^{79} - q^{81} - 6q^{83} + 12q^{85} + 3q^{87} - 3q^{89} - 2q^{93} + 8q^{95} + 20q^{97} - 24q^{99} + O(q^{100})$$

## 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.f 2
7.b odd 2 1 140.2.i.a 2
7.c even 3 1 980.2.a.e 1
7.c even 3 1 inner 980.2.i.f 2
7.d odd 6 1 140.2.i.a 2
7.d odd 6 1 980.2.a.g 1
21.c even 2 1 1260.2.s.c 2
21.g even 6 1 1260.2.s.c 2
21.g even 6 1 8820.2.a.p 1
21.h odd 6 1 8820.2.a.a 1
28.d even 2 1 560.2.q.f 2
28.f even 6 1 560.2.q.f 2
28.f even 6 1 3920.2.a.k 1
28.g odd 6 1 3920.2.a.w 1
35.c odd 2 1 700.2.i.b 2
35.f even 4 2 700.2.r.a 4
35.i odd 6 1 700.2.i.b 2
35.i odd 6 1 4900.2.a.i 1
35.j even 6 1 4900.2.a.q 1
35.k even 12 2 700.2.r.a 4
35.k even 12 2 4900.2.e.m 2
35.l odd 12 2 4900.2.e.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.a 2 7.b odd 2 1
140.2.i.a 2 7.d odd 6 1
560.2.q.f 2 28.d even 2 1
560.2.q.f 2 28.f even 6 1
700.2.i.b 2 35.c odd 2 1
700.2.i.b 2 35.i odd 6 1
700.2.r.a 4 35.f even 4 2
700.2.r.a 4 35.k even 12 2
980.2.a.e 1 7.c even 3 1
980.2.a.g 1 7.d odd 6 1
980.2.i.f 2 1.a even 1 1 trivial
980.2.i.f 2 7.c even 3 1 inner
1260.2.s.c 2 21.c even 2 1
1260.2.s.c 2 21.g even 6 1
3920.2.a.k 1 28.f even 6 1
3920.2.a.w 1 28.g odd 6 1
4900.2.a.i 1 35.i odd 6 1
4900.2.a.q 1 35.j even 6 1
4900.2.e.m 2 35.k even 12 2
4900.2.e.n 2 35.l odd 12 2
8820.2.a.a 1 21.h odd 6 1
8820.2.a.p 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 $$S_{2}^{\mathrm{new}}(980, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{11}^{2} + 6 T_{11} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$36 + 6 T + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$64 - 8 T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$( -3 + T )^{2}$$
$31$ $$4 - 2 T + T^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$( -3 + T )^{2}$$
$43$ $$( -5 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$144 + 12 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$1 + T + T^{2}$$
$67$ $$49 - 7 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$100 + 10 T + T^{2}$$
$79$ $$16 - 4 T + T^{2}$$
$83$ $$( 3 + T )^{2}$$
$89$ $$9 + 3 T + T^{2}$$
$97$ $$( -10 + T )^{2}$$