Properties

 Label 9800.2.a.bq 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 200) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} + 6q^{9} + O(q^{10})$$ $$q + 3q^{3} + 6q^{9} + q^{11} - 4q^{13} - 5q^{17} - q^{19} - 2q^{23} + 9q^{27} - 8q^{29} - 10q^{31} + 3q^{33} - 6q^{37} - 12q^{39} + 3q^{41} + 4q^{43} - 4q^{47} - 15q^{51} + 6q^{53} - 3q^{57} - 8q^{59} - 10q^{61} - q^{67} - 6q^{69} - 12q^{71} - 3q^{73} + 6q^{79} + 9q^{81} + 13q^{83} - 24q^{87} + 9q^{89} - 30q^{93} + 14q^{97} + 6q^{99} + O(q^{100})$$

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.bq 1
5.b even 2 1 9800.2.a.c 1
7.b odd 2 1 200.2.a.a 1
21.c even 2 1 1800.2.a.r 1
28.d even 2 1 400.2.a.h 1
35.c odd 2 1 200.2.a.e yes 1
35.f even 4 2 200.2.c.a 2
56.e even 2 1 1600.2.a.b 1
56.h odd 2 1 1600.2.a.x 1
84.h odd 2 1 3600.2.a.m 1
105.g even 2 1 1800.2.a.h 1
105.k odd 4 2 1800.2.f.f 2
140.c even 2 1 400.2.a.a 1
140.j odd 4 2 400.2.c.a 2
280.c odd 2 1 1600.2.a.a 1
280.n even 2 1 1600.2.a.y 1
280.s even 4 2 1600.2.c.a 2
280.y odd 4 2 1600.2.c.b 2
420.o odd 2 1 3600.2.a.bf 1
420.w even 4 2 3600.2.f.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.a.a 1 7.b odd 2 1
200.2.a.e yes 1 35.c odd 2 1
200.2.c.a 2 35.f even 4 2
400.2.a.a 1 140.c even 2 1
400.2.a.h 1 28.d even 2 1
400.2.c.a 2 140.j odd 4 2
1600.2.a.a 1 280.c odd 2 1
1600.2.a.b 1 56.e even 2 1
1600.2.a.x 1 56.h odd 2 1
1600.2.a.y 1 280.n even 2 1
1600.2.c.a 2 280.s even 4 2
1600.2.c.b 2 280.y odd 4 2
1800.2.a.h 1 105.g even 2 1
1800.2.a.r 1 21.c even 2 1
1800.2.f.f 2 105.k odd 4 2
3600.2.a.m 1 84.h odd 2 1
3600.2.a.bf 1 420.o odd 2 1
3600.2.f.n 2 420.w even 4 2
9800.2.a.c 1 5.b even 2 1
9800.2.a.bq 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3} - 3$$ $$T_{11} - 1$$ $$T_{13} + 4$$ $$T_{19} + 1$$ $$T_{23} + 2$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$-1 + T$$
$13$ $$4 + T$$
$17$ $$5 + T$$
$19$ $$1 + T$$
$23$ $$2 + T$$
$29$ $$8 + T$$
$31$ $$10 + T$$
$37$ $$6 + T$$
$41$ $$-3 + T$$
$43$ $$-4 + T$$
$47$ $$4 + T$$
$53$ $$-6 + T$$
$59$ $$8 + T$$
$61$ $$10 + T$$
$67$ $$1 + T$$
$71$ $$12 + T$$
$73$ $$3 + T$$
$79$ $$-6 + T$$
$83$ $$-13 + T$$
$89$ $$-9 + T$$
$97$ $$-14 + T$$