# Properties

 Label 9800.2.a.a Level $9800$ Weight $2$ Character orbit 9800.a Self dual yes Analytic conductor $78.253$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 3q^{3} + 6q^{9} + O(q^{10})$$ $$q - 3q^{3} + 6q^{9} - 5q^{11} - 5q^{13} - 7q^{17} + 2q^{19} + 2q^{23} - 9q^{27} + 7q^{29} - 4q^{31} + 15q^{33} + 6q^{37} + 15q^{39} + 12q^{41} + 2q^{43} + q^{47} + 21q^{51} - 6q^{57} + 4q^{59} - 4q^{61} - 8q^{67} - 6q^{69} + 6q^{73} - 3q^{79} + 9q^{81} - 4q^{83} - 21q^{87} + 12q^{93} + 13q^{97} - 30q^{99} + O(q^{100})$$

## 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 0 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 9800.2.a.a 1
5.b even 2 1 1960.2.a.o 1
7.b odd 2 1 1400.2.a.n 1
20.d odd 2 1 3920.2.a.c 1
28.d even 2 1 2800.2.a.c 1
35.c odd 2 1 280.2.a.a 1
35.f even 4 2 1400.2.g.a 2
35.i odd 6 2 1960.2.q.o 2
35.j even 6 2 1960.2.q.a 2
105.g even 2 1 2520.2.a.i 1
140.c even 2 1 560.2.a.f 1
140.j odd 4 2 2800.2.g.b 2
280.c odd 2 1 2240.2.a.z 1
280.n even 2 1 2240.2.a.a 1
420.o odd 2 1 5040.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.a 1 35.c odd 2 1
560.2.a.f 1 140.c even 2 1
1400.2.a.n 1 7.b odd 2 1
1400.2.g.a 2 35.f even 4 2
1960.2.a.o 1 5.b even 2 1
1960.2.q.a 2 35.j even 6 2
1960.2.q.o 2 35.i odd 6 2
2240.2.a.a 1 280.n even 2 1
2240.2.a.z 1 280.c odd 2 1
2520.2.a.i 1 105.g even 2 1
2800.2.a.c 1 28.d even 2 1
2800.2.g.b 2 140.j odd 4 2
3920.2.a.c 1 20.d odd 2 1
5040.2.a.a 1 420.o odd 2 1
9800.2.a.a 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

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

## Hecke characteristic polynomials

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