# Properties

 Label 9800.2.a.x 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9800,2,Mod(1,9800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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 40) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 3 q^{9}+O(q^{10})$$ q - 3 * q^9 $$q - 3 q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{17} - 4 q^{19} - 4 q^{23} - 2 q^{29} + 8 q^{31} - 6 q^{37} + 6 q^{41} + 8 q^{43} + 4 q^{47} - 6 q^{53} + 4 q^{59} + 2 q^{61} - 8 q^{67} - 6 q^{73} + 9 q^{81} - 16 q^{83} + 6 q^{89} - 14 q^{97} - 12 q^{99}+O(q^{100})$$ q - 3 * q^9 + 4 * q^11 - 2 * q^13 + 2 * q^17 - 4 * q^19 - 4 * q^23 - 2 * q^29 + 8 * q^31 - 6 * q^37 + 6 * q^41 + 8 * q^43 + 4 * q^47 - 6 * q^53 + 4 * q^59 + 2 * q^61 - 8 * q^67 - 6 * q^73 + 9 * q^81 - 16 * q^83 + 6 * q^89 - 14 * q^97 - 12 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 0 0 0 0 0 0 −3.00000 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.x 1
5.b even 2 1 1960.2.a.g 1
7.b odd 2 1 200.2.a.c 1
20.d odd 2 1 3920.2.a.s 1
21.c even 2 1 1800.2.a.v 1
28.d even 2 1 400.2.a.e 1
35.c odd 2 1 40.2.a.a 1
35.f even 4 2 200.2.c.b 2
35.i odd 6 2 1960.2.q.h 2
35.j even 6 2 1960.2.q.i 2
56.e even 2 1 1600.2.a.k 1
56.h odd 2 1 1600.2.a.o 1
84.h odd 2 1 3600.2.a.h 1
105.g even 2 1 360.2.a.a 1
105.k odd 4 2 1800.2.f.a 2
140.c even 2 1 80.2.a.a 1
140.j odd 4 2 400.2.c.d 2
280.c odd 2 1 320.2.a.c 1
280.n even 2 1 320.2.a.d 1
280.s even 4 2 1600.2.c.k 2
280.y odd 4 2 1600.2.c.m 2
315.z even 6 2 3240.2.q.x 2
315.bg odd 6 2 3240.2.q.k 2
385.h even 2 1 4840.2.a.f 1
420.o odd 2 1 720.2.a.e 1
420.w even 4 2 3600.2.f.t 2
455.h odd 2 1 6760.2.a.i 1
560.be even 4 2 1280.2.d.a 2
560.bf odd 4 2 1280.2.d.j 2
840.b odd 2 1 2880.2.a.bg 1
840.u even 2 1 2880.2.a.t 1
1540.b odd 2 1 9680.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 35.c odd 2 1
80.2.a.a 1 140.c even 2 1
200.2.a.c 1 7.b odd 2 1
200.2.c.b 2 35.f even 4 2
320.2.a.c 1 280.c odd 2 1
320.2.a.d 1 280.n even 2 1
360.2.a.a 1 105.g even 2 1
400.2.a.e 1 28.d even 2 1
400.2.c.d 2 140.j odd 4 2
720.2.a.e 1 420.o odd 2 1
1280.2.d.a 2 560.be even 4 2
1280.2.d.j 2 560.bf odd 4 2
1600.2.a.k 1 56.e even 2 1
1600.2.a.o 1 56.h odd 2 1
1600.2.c.k 2 280.s even 4 2
1600.2.c.m 2 280.y odd 4 2
1800.2.a.v 1 21.c even 2 1
1800.2.f.a 2 105.k odd 4 2
1960.2.a.g 1 5.b even 2 1
1960.2.q.h 2 35.i odd 6 2
1960.2.q.i 2 35.j even 6 2
2880.2.a.t 1 840.u even 2 1
2880.2.a.bg 1 840.b odd 2 1
3240.2.q.k 2 315.bg odd 6 2
3240.2.q.x 2 315.z even 6 2
3600.2.a.h 1 84.h odd 2 1
3600.2.f.t 2 420.w even 4 2
3920.2.a.s 1 20.d odd 2 1
4840.2.a.f 1 385.h even 2 1
6760.2.a.i 1 455.h odd 2 1
9680.2.a.q 1 1540.b odd 2 1
9800.2.a.x 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3}$$ T3 $$T_{11} - 4$$ T11 - 4 $$T_{13} + 2$$ T13 + 2 $$T_{19} + 4$$ T19 + 4 $$T_{23} + 4$$ T23 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 4$$
$13$ $$T + 2$$
$17$ $$T - 2$$
$19$ $$T + 4$$
$23$ $$T + 4$$
$29$ $$T + 2$$
$31$ $$T - 8$$
$37$ $$T + 6$$
$41$ $$T - 6$$
$43$ $$T - 8$$
$47$ $$T - 4$$
$53$ $$T + 6$$
$59$ $$T - 4$$
$61$ $$T - 2$$
$67$ $$T + 8$$
$71$ $$T$$
$73$ $$T + 6$$
$79$ $$T$$
$83$ $$T + 16$$
$89$ $$T - 6$$
$97$ $$T + 14$$