Properties

 Label 980.2.a.e 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

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} - 2q^{9} + O(q^{10})$$ $$q - q^{3} + q^{5} - 2q^{9} + 6q^{11} - 2q^{13} - q^{15} + 6q^{17} - 8q^{19} + 3q^{23} + q^{25} + 5q^{27} + 3q^{29} - 2q^{31} - 6q^{33} + 8q^{37} + 2q^{39} + 3q^{41} + 5q^{43} - 2q^{45} - 6q^{51} + 12q^{53} + 6q^{55} + 8q^{57} + q^{61} - 2q^{65} - 7q^{67} - 3q^{69} + 10q^{73} - q^{75} - 4q^{79} + q^{81} - 3q^{83} + 6q^{85} - 3q^{87} + 3q^{89} + 2q^{93} - 8q^{95} + 10q^{97} - 12q^{99} + O(q^{100})$$

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 −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.e 1
3.b odd 2 1 8820.2.a.a 1
4.b odd 2 1 3920.2.a.w 1
5.b even 2 1 4900.2.a.q 1
5.c odd 4 2 4900.2.e.n 2
7.b odd 2 1 980.2.a.g 1
7.c even 3 2 980.2.i.f 2
7.d odd 6 2 140.2.i.a 2
21.c even 2 1 8820.2.a.p 1
21.g even 6 2 1260.2.s.c 2
28.d even 2 1 3920.2.a.k 1
28.f even 6 2 560.2.q.f 2
35.c odd 2 1 4900.2.a.i 1
35.f even 4 2 4900.2.e.m 2
35.i odd 6 2 700.2.i.b 2
35.k even 12 4 700.2.r.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.a 2 7.d odd 6 2
560.2.q.f 2 28.f even 6 2
700.2.i.b 2 35.i odd 6 2
700.2.r.a 4 35.k even 12 4
980.2.a.e 1 1.a even 1 1 trivial
980.2.a.g 1 7.b odd 2 1
980.2.i.f 2 7.c even 3 2
1260.2.s.c 2 21.g even 6 2
3920.2.a.k 1 28.d even 2 1
3920.2.a.w 1 4.b odd 2 1
4900.2.a.i 1 35.c odd 2 1
4900.2.a.q 1 5.b even 2 1
4900.2.e.m 2 35.f even 4 2
4900.2.e.n 2 5.c odd 4 2
8820.2.a.a 1 3.b odd 2 1
8820.2.a.p 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(980))$$:

 $$T_{3} + 1$$ $$T_{11} - 6$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$-6 + T$$
$13$ $$2 + T$$
$17$ $$-6 + T$$
$19$ $$8 + T$$
$23$ $$-3 + T$$
$29$ $$-3 + T$$
$31$ $$2 + T$$
$37$ $$-8 + T$$
$41$ $$-3 + T$$
$43$ $$-5 + T$$
$47$ $$T$$
$53$ $$-12 + T$$
$59$ $$T$$
$61$ $$-1 + T$$
$67$ $$7 + T$$
$71$ $$T$$
$73$ $$-10 + T$$
$79$ $$4 + T$$
$83$ $$3 + T$$
$89$ $$-3 + T$$
$97$ $$-10 + T$$