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

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 280) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 6 q^{9}+O(q^{10})$$ q - 3 * q^3 + 6 * q^9 $$q - 3 q^{3} + 6 q^{9} - 5 q^{11} - 5 q^{13} - 7 q^{17} + 2 q^{19} + 2 q^{23} - 9 q^{27} + 7 q^{29} - 4 q^{31} + 15 q^{33} + 6 q^{37} + 15 q^{39} + 12 q^{41} + 2 q^{43} + q^{47} + 21 q^{51} - 6 q^{57} + 4 q^{59} - 4 q^{61} - 8 q^{67} - 6 q^{69} + 6 q^{73} - 3 q^{79} + 9 q^{81} - 4 q^{83} - 21 q^{87} + 12 q^{93} + 13 q^{97} - 30 q^{99}+O(q^{100})$$ q - 3 * q^3 + 6 * q^9 - 5 * q^11 - 5 * q^13 - 7 * q^17 + 2 * q^19 + 2 * q^23 - 9 * q^27 + 7 * q^29 - 4 * q^31 + 15 * q^33 + 6 * q^37 + 15 * q^39 + 12 * q^41 + 2 * q^43 + q^47 + 21 * q^51 - 6 * q^57 + 4 * q^59 - 4 * q^61 - 8 * q^67 - 6 * q^69 + 6 * q^73 - 3 * q^79 + 9 * q^81 - 4 * q^83 - 21 * q^87 + 12 * q^93 + 13 * q^97 - 30 * 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 −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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3} + 3$$ T3 + 3 $$T_{11} + 5$$ T11 + 5 $$T_{13} + 5$$ T13 + 5 $$T_{19} - 2$$ T19 - 2 $$T_{23} - 2$$ T23 - 2

Hecke characteristic polynomials

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