# Properties

 Label 9800.2.a.bk 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 1400) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + q^{9}+O(q^{10})$$ q + 2 * q^3 + q^9 $$q + 2 q^{3} + q^{9} + q^{11} + 4 q^{13} - 6 q^{19} - 3 q^{23} - 4 q^{27} - 3 q^{29} + 2 q^{33} - 9 q^{37} + 8 q^{39} - 2 q^{41} - 9 q^{43} - 6 q^{47} - 6 q^{53} - 12 q^{57} - 8 q^{59} + 10 q^{61} + q^{67} - 6 q^{69} - 7 q^{71} - 2 q^{73} - 9 q^{79} - 11 q^{81} + 12 q^{83} - 6 q^{87} + 4 q^{89} + 16 q^{97} + q^{99}+O(q^{100})$$ q + 2 * q^3 + q^9 + q^11 + 4 * q^13 - 6 * q^19 - 3 * q^23 - 4 * q^27 - 3 * q^29 + 2 * q^33 - 9 * q^37 + 8 * q^39 - 2 * q^41 - 9 * q^43 - 6 * q^47 - 6 * q^53 - 12 * q^57 - 8 * q^59 + 10 * q^61 + q^67 - 6 * q^69 - 7 * q^71 - 2 * q^73 - 9 * q^79 - 11 * q^81 + 12 * q^83 - 6 * q^87 + 4 * q^89 + 16 * q^97 + 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 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.bk 1
5.b even 2 1 9800.2.a.h 1
7.b odd 2 1 1400.2.a.c 1
28.d even 2 1 2800.2.a.bb 1
35.c odd 2 1 1400.2.a.l yes 1
35.f even 4 2 1400.2.g.c 2
140.c even 2 1 2800.2.a.f 1
140.j odd 4 2 2800.2.g.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.a.c 1 7.b odd 2 1
1400.2.a.l yes 1 35.c odd 2 1
1400.2.g.c 2 35.f even 4 2
2800.2.a.f 1 140.c even 2 1
2800.2.a.bb 1 28.d even 2 1
2800.2.g.f 2 140.j odd 4 2
9800.2.a.h 1 5.b even 2 1
9800.2.a.bk 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{11} - 1$$ T11 - 1 $$T_{13} - 4$$ T13 - 4 $$T_{19} + 6$$ T19 + 6 $$T_{23} + 3$$ T23 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T - 4$$
$17$ $$T$$
$19$ $$T + 6$$
$23$ $$T + 3$$
$29$ $$T + 3$$
$31$ $$T$$
$37$ $$T + 9$$
$41$ $$T + 2$$
$43$ $$T + 9$$
$47$ $$T + 6$$
$53$ $$T + 6$$
$59$ $$T + 8$$
$61$ $$T - 10$$
$67$ $$T - 1$$
$71$ $$T + 7$$
$73$ $$T + 2$$
$79$ $$T + 9$$
$83$ $$T - 12$$
$89$ $$T - 4$$
$97$ $$T - 16$$