# Properties

 Label 9800.2.a.g Level $9800$ Weight $2$ Character orbit 9800.a Self dual yes Analytic conductor $78.253$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9800 = 2^{3} \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$78.2533939809$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{3} + q^{9} + O(q^{10})$$ $$q - 2q^{3} + q^{9} - q^{11} - 3q^{13} - 2q^{17} + 5q^{19} - 7q^{23} + 4q^{27} - 6q^{29} - 4q^{31} + 2q^{33} + 5q^{37} + 6q^{39} + 5q^{41} - 6q^{43} - 9q^{47} + 4q^{51} - 11q^{53} - 10q^{57} - 8q^{59} + 12q^{61} + 4q^{67} + 14q^{69} - 4q^{71} + 12q^{73} + 14q^{79} - 11q^{81} - 4q^{83} + 12q^{87} - 6q^{89} + 8q^{93} + 6q^{97} - q^{99} + O(q^{100})$$

## 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 −2.00000 0 0 0 0 0 1.00000 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 9800.2.a.g 1
5.b even 2 1 1960.2.a.m 1
7.b odd 2 1 9800.2.a.bi 1
7.d odd 6 2 1400.2.q.a 2
20.d odd 2 1 3920.2.a.i 1
35.c odd 2 1 1960.2.a.a 1
35.i odd 6 2 280.2.q.c 2
35.j even 6 2 1960.2.q.c 2
35.k even 12 4 1400.2.bh.e 4
105.p even 6 2 2520.2.bi.e 2
140.c even 2 1 3920.2.a.bf 1
140.s even 6 2 560.2.q.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.c 2 35.i odd 6 2
560.2.q.c 2 140.s even 6 2
1400.2.q.a 2 7.d odd 6 2
1400.2.bh.e 4 35.k even 12 4
1960.2.a.a 1 35.c odd 2 1
1960.2.a.m 1 5.b even 2 1
1960.2.q.c 2 35.j even 6 2
2520.2.bi.e 2 105.p even 6 2
3920.2.a.i 1 20.d odd 2 1
3920.2.a.bf 1 140.c even 2 1
9800.2.a.g 1 1.a even 1 1 trivial
9800.2.a.bi 1 7.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3} + 2$$ $$T_{11} + 1$$ $$T_{13} + 3$$ $$T_{19} - 5$$ $$T_{23} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$2 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$1 + T$$
$13$ $$3 + T$$
$17$ $$2 + T$$
$19$ $$-5 + T$$
$23$ $$7 + T$$
$29$ $$6 + T$$
$31$ $$4 + T$$
$37$ $$-5 + T$$
$41$ $$-5 + T$$
$43$ $$6 + T$$
$47$ $$9 + T$$
$53$ $$11 + T$$
$59$ $$8 + T$$
$61$ $$-12 + T$$
$67$ $$-4 + T$$
$71$ $$4 + T$$
$73$ $$-12 + T$$
$79$ $$-14 + T$$
$83$ $$4 + T$$
$89$ $$6 + T$$
$97$ $$-6 + T$$