# Properties

 Label 9800.2.a.s 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 56) 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 $$q - q^{3} - 2 q^{9} + 3 q^{11} - 6 q^{13} - 5 q^{17} - q^{19} + 7 q^{23} + 5 q^{27} + 2 q^{29} + 5 q^{31} - 3 q^{33} - 3 q^{37} + 6 q^{39} + 2 q^{41} + 4 q^{43} + 5 q^{47} + 5 q^{51} + q^{53} + q^{57} - 15 q^{59} + 5 q^{61} + 9 q^{67} - 7 q^{69} + 7 q^{73} + q^{79} + q^{81} + 12 q^{83} - 2 q^{87} - 7 q^{89} - 5 q^{93} - 2 q^{97} - 6 q^{99}+O(q^{100})$$ q - q^3 - 2 * q^9 + 3 * q^11 - 6 * q^13 - 5 * q^17 - q^19 + 7 * q^23 + 5 * q^27 + 2 * q^29 + 5 * q^31 - 3 * q^33 - 3 * q^37 + 6 * q^39 + 2 * q^41 + 4 * q^43 + 5 * q^47 + 5 * q^51 + q^53 + q^57 - 15 * q^59 + 5 * q^61 + 9 * q^67 - 7 * q^69 + 7 * q^73 + q^79 + q^81 + 12 * q^83 - 2 * q^87 - 7 * q^89 - 5 * q^93 - 2 * q^97 - 6 * 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.s 1
5.b even 2 1 392.2.a.e 1
7.b odd 2 1 9800.2.a.be 1
7.d odd 6 2 1400.2.q.d 2
15.d odd 2 1 3528.2.a.j 1
20.d odd 2 1 784.2.a.c 1
35.c odd 2 1 392.2.a.c 1
35.i odd 6 2 56.2.i.b 2
35.j even 6 2 392.2.i.b 2
35.k even 12 4 1400.2.bh.a 4
40.e odd 2 1 3136.2.a.t 1
40.f even 2 1 3136.2.a.i 1
60.h even 2 1 7056.2.a.u 1
105.g even 2 1 3528.2.a.p 1
105.o odd 6 2 3528.2.s.q 2
105.p even 6 2 504.2.s.c 2
140.c even 2 1 784.2.a.h 1
140.p odd 6 2 784.2.i.h 2
140.s even 6 2 112.2.i.a 2
280.c odd 2 1 3136.2.a.u 1
280.n even 2 1 3136.2.a.j 1
280.ba even 6 2 448.2.i.d 2
280.bk odd 6 2 448.2.i.b 2
420.o odd 2 1 7056.2.a.bj 1
420.be odd 6 2 1008.2.s.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.b 2 35.i odd 6 2
112.2.i.a 2 140.s even 6 2
392.2.a.c 1 35.c odd 2 1
392.2.a.e 1 5.b even 2 1
392.2.i.b 2 35.j even 6 2
448.2.i.b 2 280.bk odd 6 2
448.2.i.d 2 280.ba even 6 2
504.2.s.c 2 105.p even 6 2
784.2.a.c 1 20.d odd 2 1
784.2.a.h 1 140.c even 2 1
784.2.i.h 2 140.p odd 6 2
1008.2.s.g 2 420.be odd 6 2
1400.2.q.d 2 7.d odd 6 2
1400.2.bh.a 4 35.k even 12 4
3136.2.a.i 1 40.f even 2 1
3136.2.a.j 1 280.n even 2 1
3136.2.a.t 1 40.e odd 2 1
3136.2.a.u 1 280.c odd 2 1
3528.2.a.j 1 15.d odd 2 1
3528.2.a.p 1 105.g even 2 1
3528.2.s.q 2 105.o odd 6 2
7056.2.a.u 1 60.h even 2 1
7056.2.a.bj 1 420.o odd 2 1
9800.2.a.s 1 1.a even 1 1 trivial
9800.2.a.be 1 7.b odd 2 1

## 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 $$T_{11} - 3$$ T11 - 3 $$T_{13} + 6$$ T13 + 6 $$T_{19} + 1$$ T19 + 1 $$T_{23} - 7$$ T23 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 3$$
$13$ $$T + 6$$
$17$ $$T + 5$$
$19$ $$T + 1$$
$23$ $$T - 7$$
$29$ $$T - 2$$
$31$ $$T - 5$$
$37$ $$T + 3$$
$41$ $$T - 2$$
$43$ $$T - 4$$
$47$ $$T - 5$$
$53$ $$T - 1$$
$59$ $$T + 15$$
$61$ $$T - 5$$
$67$ $$T - 9$$
$71$ $$T$$
$73$ $$T - 7$$
$79$ $$T - 1$$
$83$ $$T - 12$$
$89$ $$T + 7$$
$97$ $$T + 2$$