# Properties

 Label 980.2.i.d 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} - q^{13} + q^{15} + ( 3 - 3 \zeta_{6} ) q^{17} -2 \zeta_{6} q^{19} + 6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -5 q^{27} -9 q^{29} + ( -8 + 8 \zeta_{6} ) q^{31} -3 \zeta_{6} q^{33} + 10 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{39} + 2 q^{43} + ( 2 - 2 \zeta_{6} ) q^{45} + 3 \zeta_{6} q^{47} + 3 \zeta_{6} q^{51} + 3 q^{55} + 2 q^{57} + ( -12 + 12 \zeta_{6} ) q^{59} -8 \zeta_{6} q^{61} + \zeta_{6} q^{65} + ( -8 + 8 \zeta_{6} ) q^{67} -6 q^{69} + ( -14 + 14 \zeta_{6} ) q^{73} -\zeta_{6} q^{75} -5 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -12 q^{83} -3 q^{85} + ( 9 - 9 \zeta_{6} ) q^{87} -12 \zeta_{6} q^{89} -8 \zeta_{6} q^{93} + ( -2 + 2 \zeta_{6} ) q^{95} + 17 q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} - q^{5} + 2q^{9} + O(q^{10})$$ $$2q - q^{3} - q^{5} + 2q^{9} - 3q^{11} - 2q^{13} + 2q^{15} + 3q^{17} - 2q^{19} + 6q^{23} - q^{25} - 10q^{27} - 18q^{29} - 8q^{31} - 3q^{33} + 10q^{37} + q^{39} + 4q^{43} + 2q^{45} + 3q^{47} + 3q^{51} + 6q^{55} + 4q^{57} - 12q^{59} - 8q^{61} + q^{65} - 8q^{67} - 12q^{69} - 14q^{73} - q^{75} - 5q^{79} - q^{81} - 24q^{83} - 6q^{85} + 9q^{87} - 12q^{89} - 8q^{93} - 2q^{95} + 34q^{97} - 12q^{99} + O(q^{100})$$

## 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.d 2
7.b odd 2 1 980.2.i.h 2
7.c even 3 1 140.2.a.a 1
7.c even 3 1 inner 980.2.i.d 2
7.d odd 6 1 980.2.a.c 1
7.d odd 6 1 980.2.i.h 2
21.g even 6 1 8820.2.a.r 1
21.h odd 6 1 1260.2.a.c 1
28.f even 6 1 3920.2.a.u 1
28.g odd 6 1 560.2.a.c 1
35.i odd 6 1 4900.2.a.p 1
35.j even 6 1 700.2.a.d 1
35.k even 12 2 4900.2.e.l 2
35.l odd 12 2 700.2.e.c 2
56.k odd 6 1 2240.2.a.r 1
56.p even 6 1 2240.2.a.g 1
84.n even 6 1 5040.2.a.h 1
105.o odd 6 1 6300.2.a.d 1
105.x even 12 2 6300.2.k.c 2
140.p odd 6 1 2800.2.a.y 1
140.w even 12 2 2800.2.g.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.a 1 7.c even 3 1
560.2.a.c 1 28.g odd 6 1
700.2.a.d 1 35.j even 6 1
700.2.e.c 2 35.l odd 12 2
980.2.a.c 1 7.d odd 6 1
980.2.i.d 2 1.a even 1 1 trivial
980.2.i.d 2 7.c even 3 1 inner
980.2.i.h 2 7.b odd 2 1
980.2.i.h 2 7.d odd 6 1
1260.2.a.c 1 21.h odd 6 1
2240.2.a.g 1 56.p even 6 1
2240.2.a.r 1 56.k odd 6 1
2800.2.a.y 1 140.p odd 6 1
2800.2.g.j 2 140.w even 12 2
3920.2.a.u 1 28.f even 6 1
4900.2.a.p 1 35.i odd 6 1
4900.2.e.l 2 35.k even 12 2
5040.2.a.h 1 84.n even 6 1
6300.2.a.d 1 105.o odd 6 1
6300.2.k.c 2 105.x even 12 2
8820.2.a.r 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 $$S_{2}^{\mathrm{new}}(980, [\chi])$$:

 $$T_{3}^{2} + T_{3} + 1$$ $$T_{11}^{2} + 3 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$64 + 8 T + T^{2}$$
$37$ $$100 - 10 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$9 - 3 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$64 + 8 T + T^{2}$$
$67$ $$64 + 8 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$196 + 14 T + T^{2}$$
$79$ $$25 + 5 T + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$144 + 12 T + T^{2}$$
$97$ $$( -17 + T )^{2}$$