# Properties

 Label 9702.2.a.ce Level $9702$ Weight $2$ Character orbit 9702.a Self dual yes Analytic conductor $77.471$ 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] = [9702,2,Mod(1,9702)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9702, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9702.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9702.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: $$77.4708600410$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 462) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 3 q^{5} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + 3 * q^5 + q^8 $$q + q^{2} + q^{4} + 3 q^{5} + q^{8} + 3 q^{10} + q^{11} - 6 q^{13} + q^{16} - 5 q^{17} - 6 q^{19} + 3 q^{20} + q^{22} - 5 q^{23} + 4 q^{25} - 6 q^{26} + 6 q^{29} - 4 q^{31} + q^{32} - 5 q^{34} - 2 q^{37} - 6 q^{38} + 3 q^{40} + 5 q^{41} - 10 q^{43} + q^{44} - 5 q^{46} + 9 q^{47} + 4 q^{50} - 6 q^{52} - 2 q^{53} + 3 q^{55} + 6 q^{58} - 12 q^{59} + 5 q^{61} - 4 q^{62} + q^{64} - 18 q^{65} + 5 q^{67} - 5 q^{68} - 4 q^{71} - 12 q^{73} - 2 q^{74} - 6 q^{76} - q^{79} + 3 q^{80} + 5 q^{82} + q^{83} - 15 q^{85} - 10 q^{86} + q^{88} + 6 q^{89} - 5 q^{92} + 9 q^{94} - 18 q^{95} - 9 q^{97}+O(q^{100})$$ q + q^2 + q^4 + 3 * q^5 + q^8 + 3 * q^10 + q^11 - 6 * q^13 + q^16 - 5 * q^17 - 6 * q^19 + 3 * q^20 + q^22 - 5 * q^23 + 4 * q^25 - 6 * q^26 + 6 * q^29 - 4 * q^31 + q^32 - 5 * q^34 - 2 * q^37 - 6 * q^38 + 3 * q^40 + 5 * q^41 - 10 * q^43 + q^44 - 5 * q^46 + 9 * q^47 + 4 * q^50 - 6 * q^52 - 2 * q^53 + 3 * q^55 + 6 * q^58 - 12 * q^59 + 5 * q^61 - 4 * q^62 + q^64 - 18 * q^65 + 5 * q^67 - 5 * q^68 - 4 * q^71 - 12 * q^73 - 2 * q^74 - 6 * q^76 - q^79 + 3 * q^80 + 5 * q^82 + q^83 - 15 * q^85 - 10 * q^86 + q^88 + 6 * q^89 - 5 * q^92 + 9 * q^94 - 18 * q^95 - 9 * q^97

## 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
1.00000 0 1.00000 3.00000 0 0 1.00000 0 3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 9702.2.a.ce 1
3.b odd 2 1 3234.2.a.a 1
7.b odd 2 1 9702.2.a.be 1
7.d odd 6 2 1386.2.k.j 2
21.c even 2 1 3234.2.a.o 1
21.g even 6 2 462.2.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.a 2 21.g even 6 2
1386.2.k.j 2 7.d odd 6 2
3234.2.a.a 1 3.b odd 2 1
3234.2.a.o 1 21.c even 2 1
9702.2.a.be 1 7.b odd 2 1
9702.2.a.ce 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9702))$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{13} + 6$$ T13 + 6 $$T_{17} + 5$$ T17 + 5 $$T_{19} + 6$$ T19 + 6 $$T_{23} + 5$$ T23 + 5 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

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