# Properties

 Label 980.2.e.b Level $980$ Weight $2$ Character orbit 980.e 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.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82533939809$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + ( 2 + i ) q^{5} -6 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} + ( 2 + i ) q^{5} -6 q^{9} + 3 q^{11} + i q^{13} + ( -3 + 6 i ) q^{15} + 5 i q^{17} -8 q^{19} -2 i q^{23} + ( 3 + 4 i ) q^{25} -9 i q^{27} + q^{29} + 2 q^{31} + 9 i q^{33} + 10 i q^{37} -3 q^{39} + 6 q^{41} + 4 i q^{43} + ( -12 - 6 i ) q^{45} -11 i q^{47} -15 q^{51} -6 i q^{53} + ( 6 + 3 i ) q^{55} -24 i q^{57} -10 q^{59} + ( -1 + 2 i ) q^{65} -10 i q^{67} + 6 q^{69} -10 i q^{73} + ( -12 + 9 i ) q^{75} + 7 q^{79} + 9 q^{81} + 12 i q^{83} + ( -5 + 10 i ) q^{85} + 3 i q^{87} + 8 q^{89} + 6 i q^{93} + ( -16 - 8 i ) q^{95} -3 i q^{97} -18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} - 12 q^{9} + O(q^{10})$$ $$2 q + 4 q^{5} - 12 q^{9} + 6 q^{11} - 6 q^{15} - 16 q^{19} + 6 q^{25} + 2 q^{29} + 4 q^{31} - 6 q^{39} + 12 q^{41} - 24 q^{45} - 30 q^{51} + 12 q^{55} - 20 q^{59} - 2 q^{65} + 12 q^{69} - 24 q^{75} + 14 q^{79} + 18 q^{81} - 10 q^{85} + 16 q^{89} - 32 q^{95} - 36 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
589.1
 − 1.00000i 1.00000i
0 3.00000i 0 2.00000 1.00000i 0 0 0 −6.00000 0
589.2 0 3.00000i 0 2.00000 + 1.00000i 0 0 0 −6.00000 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 980.2.e.b 2
5.b even 2 1 inner 980.2.e.b 2
5.c odd 4 1 4900.2.a.b 1
5.c odd 4 1 4900.2.a.w 1
7.b odd 2 1 140.2.e.a 2
7.c even 3 2 980.2.q.c 4
7.d odd 6 2 980.2.q.f 4
21.c even 2 1 1260.2.k.c 2
28.d even 2 1 560.2.g.a 2
35.c odd 2 1 140.2.e.a 2
35.f even 4 1 700.2.a.a 1
35.f even 4 1 700.2.a.j 1
35.i odd 6 2 980.2.q.f 4
35.j even 6 2 980.2.q.c 4
56.e even 2 1 2240.2.g.f 2
56.h odd 2 1 2240.2.g.e 2
84.h odd 2 1 5040.2.t.s 2
105.g even 2 1 1260.2.k.c 2
105.k odd 4 1 6300.2.a.c 1
105.k odd 4 1 6300.2.a.t 1
140.c even 2 1 560.2.g.a 2
140.j odd 4 1 2800.2.a.a 1
140.j odd 4 1 2800.2.a.bf 1
280.c odd 2 1 2240.2.g.e 2
280.n even 2 1 2240.2.g.f 2
420.o odd 2 1 5040.2.t.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 7.b odd 2 1
140.2.e.a 2 35.c odd 2 1
560.2.g.a 2 28.d even 2 1
560.2.g.a 2 140.c even 2 1
700.2.a.a 1 35.f even 4 1
700.2.a.j 1 35.f even 4 1
980.2.e.b 2 1.a even 1 1 trivial
980.2.e.b 2 5.b even 2 1 inner
980.2.q.c 4 7.c even 3 2
980.2.q.c 4 35.j even 6 2
980.2.q.f 4 7.d odd 6 2
980.2.q.f 4 35.i odd 6 2
1260.2.k.c 2 21.c even 2 1
1260.2.k.c 2 105.g even 2 1
2240.2.g.e 2 56.h odd 2 1
2240.2.g.e 2 280.c odd 2 1
2240.2.g.f 2 56.e even 2 1
2240.2.g.f 2 280.n even 2 1
2800.2.a.a 1 140.j odd 4 1
2800.2.a.bf 1 140.j odd 4 1
4900.2.a.b 1 5.c odd 4 1
4900.2.a.w 1 5.c odd 4 1
5040.2.t.s 2 84.h odd 2 1
5040.2.t.s 2 420.o odd 2 1
6300.2.a.c 1 105.k odd 4 1
6300.2.a.t 1 105.k odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 9$$ $$T_{11} - 3$$ $$T_{19} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$5 - 4 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$25 + T^{2}$$
$19$ $$( 8 + T )^{2}$$
$23$ $$4 + T^{2}$$
$29$ $$( -1 + T )^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$121 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 10 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$100 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( -7 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -8 + T )^{2}$$
$97$ $$9 + T^{2}$$