# Properties

 Label 980.2.a.i Level $980$ Weight $2$ Character orbit 980.a Self dual yes Analytic conductor $7.825$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 980.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.82533939809$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} + q^{5} + 6q^{9} + O(q^{10})$$ $$q + 3q^{3} + q^{5} + 6q^{9} - 2q^{11} + 6q^{13} + 3q^{15} - 2q^{17} - 9q^{23} + q^{25} + 9q^{27} + 3q^{29} - 2q^{31} - 6q^{33} + 8q^{37} + 18q^{39} - 5q^{41} + q^{43} + 6q^{45} - 8q^{47} - 6q^{51} + 4q^{53} - 2q^{55} + 8q^{59} - 7q^{61} + 6q^{65} - 3q^{67} - 27q^{69} + 8q^{71} - 14q^{73} + 3q^{75} + 4q^{79} + 9q^{81} + q^{83} - 2q^{85} + 9q^{87} - 13q^{89} - 6q^{93} + 10q^{97} - 12q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 3.00000 0 1.00000 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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.a.i 1
3.b odd 2 1 8820.2.a.k 1
4.b odd 2 1 3920.2.a.d 1
5.b even 2 1 4900.2.a.a 1
5.c odd 4 2 4900.2.e.b 2
7.b odd 2 1 980.2.a.a 1
7.c even 3 2 980.2.i.a 2
7.d odd 6 2 140.2.i.b 2
21.c even 2 1 8820.2.a.w 1
21.g even 6 2 1260.2.s.b 2
28.d even 2 1 3920.2.a.bi 1
28.f even 6 2 560.2.q.a 2
35.c odd 2 1 4900.2.a.v 1
35.f even 4 2 4900.2.e.c 2
35.i odd 6 2 700.2.i.a 2
35.k even 12 4 700.2.r.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.b 2 7.d odd 6 2
560.2.q.a 2 28.f even 6 2
700.2.i.a 2 35.i odd 6 2
700.2.r.c 4 35.k even 12 4
980.2.a.a 1 7.b odd 2 1
980.2.a.i 1 1.a even 1 1 trivial
980.2.i.a 2 7.c even 3 2
1260.2.s.b 2 21.g even 6 2
3920.2.a.d 1 4.b odd 2 1
3920.2.a.bi 1 28.d even 2 1
4900.2.a.a 1 5.b even 2 1
4900.2.a.v 1 35.c odd 2 1
4900.2.e.b 2 5.c odd 4 2
4900.2.e.c 2 35.f even 4 2
8820.2.a.k 1 3.b odd 2 1
8820.2.a.w 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(980))$$:

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

## Hecke characteristic polynomials

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