# Properties

 Label 980.2.e.a 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: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{5} + 3 q^{9}+O(q^{10})$$ q + (-b - 1) * q^5 + 3 * q^9 $$q + ( - \beta - 1) q^{5} + 3 q^{9} + 2 \beta q^{13} - 2 \beta q^{17} + 4 q^{19} - 4 \beta q^{23} + (2 \beta - 3) q^{25} - 2 q^{29} + 8 q^{31} - 4 \beta q^{37} - 6 q^{41} - 4 \beta q^{43} + ( - 3 \beta - 3) q^{45} - 4 \beta q^{47} - 4 q^{59} + 6 q^{61} + ( - 2 \beta + 8) q^{65} + 4 \beta q^{67} + 12 q^{71} - 2 \beta q^{73} + 4 q^{79} + 9 q^{81} + (2 \beta - 8) q^{85} - 10 q^{89} + ( - 4 \beta - 4) q^{95} + 6 \beta q^{97} +O(q^{100})$$ q + (-b - 1) * q^5 + 3 * q^9 + 2*b * q^13 - 2*b * q^17 + 4 * q^19 - 4*b * q^23 + (2*b - 3) * q^25 - 2 * q^29 + 8 * q^31 - 4*b * q^37 - 6 * q^41 - 4*b * q^43 + (-3*b - 3) * q^45 - 4*b * q^47 - 4 * q^59 + 6 * q^61 + (-2*b + 8) * q^65 + 4*b * q^67 + 12 * q^71 - 2*b * q^73 + 4 * q^79 + 9 * q^81 + (2*b - 8) * q^85 - 10 * q^89 + (-4*b - 4) * q^95 + 6*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 + 6 * q^9 $$2 q - 2 q^{5} + 6 q^{9} + 8 q^{19} - 6 q^{25} - 4 q^{29} + 16 q^{31} - 12 q^{41} - 6 q^{45} - 8 q^{59} + 12 q^{61} + 16 q^{65} + 24 q^{71} + 8 q^{79} + 18 q^{81} - 16 q^{85} - 20 q^{89} - 8 q^{95}+O(q^{100})$$ 2 * q - 2 * q^5 + 6 * q^9 + 8 * q^19 - 6 * q^25 - 4 * q^29 + 16 * q^31 - 12 * q^41 - 6 * q^45 - 8 * q^59 + 12 * q^61 + 16 * q^65 + 24 * q^71 + 8 * q^79 + 18 * q^81 - 16 * q^85 - 20 * q^89 - 8 * q^95

## 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 0 0 −1.00000 2.00000i 0 0 0 3.00000 0
589.2 0 0 0 −1.00000 + 2.00000i 0 0 0 3.00000 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.a 2
5.b even 2 1 inner 980.2.e.a 2
5.c odd 4 1 4900.2.a.l 1
5.c odd 4 1 4900.2.a.m 1
7.b odd 2 1 140.2.e.b 2
7.c even 3 2 980.2.q.e 4
7.d odd 6 2 980.2.q.d 4
21.c even 2 1 1260.2.k.b 2
28.d even 2 1 560.2.g.c 2
35.c odd 2 1 140.2.e.b 2
35.f even 4 1 700.2.a.f 1
35.f even 4 1 700.2.a.h 1
35.i odd 6 2 980.2.q.d 4
35.j even 6 2 980.2.q.e 4
56.e even 2 1 2240.2.g.c 2
56.h odd 2 1 2240.2.g.d 2
84.h odd 2 1 5040.2.t.g 2
105.g even 2 1 1260.2.k.b 2
105.k odd 4 1 6300.2.a.g 1
105.k odd 4 1 6300.2.a.y 1
140.c even 2 1 560.2.g.c 2
140.j odd 4 1 2800.2.a.o 1
140.j odd 4 1 2800.2.a.s 1
280.c odd 2 1 2240.2.g.d 2
280.n even 2 1 2240.2.g.c 2
420.o odd 2 1 5040.2.t.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.b 2 7.b odd 2 1
140.2.e.b 2 35.c odd 2 1
560.2.g.c 2 28.d even 2 1
560.2.g.c 2 140.c even 2 1
700.2.a.f 1 35.f even 4 1
700.2.a.h 1 35.f even 4 1
980.2.e.a 2 1.a even 1 1 trivial
980.2.e.a 2 5.b even 2 1 inner
980.2.q.d 4 7.d odd 6 2
980.2.q.d 4 35.i odd 6 2
980.2.q.e 4 7.c even 3 2
980.2.q.e 4 35.j even 6 2
1260.2.k.b 2 21.c even 2 1
1260.2.k.b 2 105.g even 2 1
2240.2.g.c 2 56.e even 2 1
2240.2.g.c 2 280.n even 2 1
2240.2.g.d 2 56.h odd 2 1
2240.2.g.d 2 280.c odd 2 1
2800.2.a.o 1 140.j odd 4 1
2800.2.a.s 1 140.j odd 4 1
4900.2.a.l 1 5.c odd 4 1
4900.2.a.m 1 5.c odd 4 1
5040.2.t.g 2 84.h odd 2 1
5040.2.t.g 2 420.o odd 2 1
6300.2.a.g 1 105.k odd 4 1
6300.2.a.y 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 $$S_{2}^{\mathrm{new}}(980, [\chi])$$:

 $$T_{3}$$ T3 $$T_{11}$$ T11 $$T_{19} - 4$$ T19 - 4

## Hecke characteristic polynomials

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