# Properties

 Label 980.2.a.f Level $980$ Weight $2$ Character orbit 980.a Self dual yes Analytic conductor $7.825$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} - 2q^{9} + O(q^{10})$$ $$q + q^{3} - q^{5} - 2q^{9} - q^{11} - 5q^{13} - q^{15} + q^{17} - 6q^{19} - 4q^{23} + q^{25} - 5q^{27} + 3q^{29} + 2q^{31} - q^{33} + 8q^{37} - 5q^{39} - 10q^{41} - 2q^{43} + 2q^{45} - 7q^{47} + q^{51} - 2q^{53} + q^{55} - 6q^{57} + 14q^{59} - 8q^{61} + 5q^{65} + 14q^{67} - 4q^{69} - 10q^{73} + q^{75} - 11q^{79} + q^{81} - 4q^{83} - q^{85} + 3q^{87} + 4q^{89} + 2q^{93} + 6q^{95} - 3q^{97} + 2q^{99} + O(q^{100})$$

## 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 1.00000 0 −1.00000 0 0 0 −2.00000 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.f yes 1
3.b odd 2 1 8820.2.a.v 1
4.b odd 2 1 3920.2.a.n 1
5.b even 2 1 4900.2.a.h 1
5.c odd 4 2 4900.2.e.k 2
7.b odd 2 1 980.2.a.d 1
7.c even 3 2 980.2.i.e 2
7.d odd 6 2 980.2.i.g 2
21.c even 2 1 8820.2.a.i 1
28.d even 2 1 3920.2.a.z 1
35.c odd 2 1 4900.2.a.o 1
35.f even 4 2 4900.2.e.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.d 1 7.b odd 2 1
980.2.a.f yes 1 1.a even 1 1 trivial
980.2.i.e 2 7.c even 3 2
980.2.i.g 2 7.d odd 6 2
3920.2.a.n 1 4.b odd 2 1
3920.2.a.z 1 28.d even 2 1
4900.2.a.h 1 5.b even 2 1
4900.2.a.o 1 35.c odd 2 1
4900.2.e.j 2 35.f even 4 2
4900.2.e.k 2 5.c odd 4 2
8820.2.a.i 1 21.c even 2 1
8820.2.a.v 1 3.b odd 2 1

## Hecke kernels

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

 $$T_{3} - 1$$ $$T_{11} + 1$$

## Hecke characteristic polynomials

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