# Properties

 Label 9800.2.a.ba 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

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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{9}+O(q^{10})$$ q + q^3 - 2 * q^9 $$q + q^{3} - 2 q^{9} - q^{11} - 6 q^{13} + 7 q^{17} - q^{19} + 8 q^{23} - 5 q^{27} - 6 q^{29} - 4 q^{31} - q^{33} + 8 q^{37} - 6 q^{39} + 5 q^{41} - 6 q^{47} + 7 q^{51} + 4 q^{53} - q^{57} + 4 q^{59} - 6 q^{61} - 5 q^{67} + 8 q^{69} + 14 q^{71} - 15 q^{73} + 14 q^{79} + q^{81} - q^{83} - 6 q^{87} + 3 q^{89} - 4 q^{93} + 6 q^{97} + 2 q^{99}+O(q^{100})$$ q + q^3 - 2 * q^9 - q^11 - 6 * q^13 + 7 * q^17 - q^19 + 8 * q^23 - 5 * q^27 - 6 * q^29 - 4 * q^31 - q^33 + 8 * q^37 - 6 * q^39 + 5 * q^41 - 6 * q^47 + 7 * q^51 + 4 * q^53 - q^57 + 4 * q^59 - 6 * q^61 - 5 * q^67 + 8 * q^69 + 14 * q^71 - 15 * q^73 + 14 * q^79 + q^81 - q^83 - 6 * q^87 + 3 * q^89 - 4 * q^93 + 6 * q^97 + 2 * 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 1.00000 0 0 0 0 0 −2.00000 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.ba 1
5.b even 2 1 9800.2.a.q 1
7.b odd 2 1 1400.2.a.e 1
28.d even 2 1 2800.2.a.v 1
35.c odd 2 1 1400.2.a.i yes 1
35.f even 4 2 1400.2.g.f 2
140.c even 2 1 2800.2.a.j 1
140.j odd 4 2 2800.2.g.k 2

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

 $$T_{3} - 1$$ T3 - 1 $$T_{11} + 1$$ T11 + 1 $$T_{13} + 6$$ T13 + 6 $$T_{19} + 1$$ T19 + 1 $$T_{23} - 8$$ T23 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T + 6$$
$17$ $$T - 7$$
$19$ $$T + 1$$
$23$ $$T - 8$$
$29$ $$T + 6$$
$31$ $$T + 4$$
$37$ $$T - 8$$
$41$ $$T - 5$$
$43$ $$T$$
$47$ $$T + 6$$
$53$ $$T - 4$$
$59$ $$T - 4$$
$61$ $$T + 6$$
$67$ $$T + 5$$
$71$ $$T - 14$$
$73$ $$T + 15$$
$79$ $$T - 14$$
$83$ $$T + 1$$
$89$ $$T - 3$$
$97$ $$T - 6$$