# Properties

 Label 9702.2.a.i 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 154) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - 2 q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 - 2 * q^5 - q^8 $$q - q^{2} + q^{4} - 2 q^{5} - q^{8} + 2 q^{10} + q^{11} - 7 q^{13} + q^{16} - 2 q^{17} - 2 q^{20} - q^{22} + 8 q^{23} - q^{25} + 7 q^{26} + 5 q^{29} + 4 q^{31} - q^{32} + 2 q^{34} + 4 q^{37} + 2 q^{40} - 4 q^{41} - 8 q^{43} + q^{44} - 8 q^{46} - 2 q^{47} + q^{50} - 7 q^{52} + 6 q^{53} - 2 q^{55} - 5 q^{58} - 3 q^{59} + q^{61} - 4 q^{62} + q^{64} + 14 q^{65} + 9 q^{67} - 2 q^{68} + 2 q^{71} + 4 q^{73} - 4 q^{74} + 9 q^{79} - 2 q^{80} + 4 q^{82} - 6 q^{83} + 4 q^{85} + 8 q^{86} - q^{88} - 6 q^{89} + 8 q^{92} + 2 q^{94} + 7 q^{97}+O(q^{100})$$ q - q^2 + q^4 - 2 * q^5 - q^8 + 2 * q^10 + q^11 - 7 * q^13 + q^16 - 2 * q^17 - 2 * q^20 - q^22 + 8 * q^23 - q^25 + 7 * q^26 + 5 * q^29 + 4 * q^31 - q^32 + 2 * q^34 + 4 * q^37 + 2 * q^40 - 4 * q^41 - 8 * q^43 + q^44 - 8 * q^46 - 2 * q^47 + q^50 - 7 * q^52 + 6 * q^53 - 2 * q^55 - 5 * q^58 - 3 * q^59 + q^61 - 4 * q^62 + q^64 + 14 * q^65 + 9 * q^67 - 2 * q^68 + 2 * q^71 + 4 * q^73 - 4 * q^74 + 9 * q^79 - 2 * q^80 + 4 * q^82 - 6 * q^83 + 4 * q^85 + 8 * q^86 - q^88 - 6 * q^89 + 8 * q^92 + 2 * q^94 + 7 * 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 −2.00000 0 0 −1.00000 0 2.00000
 $$n$$: 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.i 1
3.b odd 2 1 1078.2.a.m 1
7.b odd 2 1 9702.2.a.y 1
7.c even 3 2 1386.2.k.o 2
12.b even 2 1 8624.2.a.b 1
21.c even 2 1 1078.2.a.g 1
21.g even 6 2 1078.2.e.f 2
21.h odd 6 2 154.2.e.a 2
84.h odd 2 1 8624.2.a.be 1
84.n even 6 2 1232.2.q.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.a 2 21.h odd 6 2
1078.2.a.g 1 21.c even 2 1
1078.2.a.m 1 3.b odd 2 1
1078.2.e.f 2 21.g even 6 2
1232.2.q.e 2 84.n even 6 2
1386.2.k.o 2 7.c even 3 2
8624.2.a.b 1 12.b even 2 1
8624.2.a.be 1 84.h odd 2 1
9702.2.a.i 1 1.a even 1 1 trivial
9702.2.a.y 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9702))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{13} + 7$$ T13 + 7 $$T_{17} + 2$$ T17 + 2 $$T_{19}$$ T19 $$T_{23} - 8$$ T23 - 8 $$T_{29} - 5$$ T29 - 5

## Hecke characteristic polynomials

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