# Properties

 Label 9702.2.a.cp Level $9702$ Weight $2$ Character orbit 9702.a Self dual yes Analytic conductor $77.471$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1078) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + 3 \beta q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + 3*b * q^5 - q^8 $$q - q^{2} + q^{4} + 3 \beta q^{5} - q^{8} - 3 \beta q^{10} + q^{11} + q^{16} - 4 \beta q^{17} + 3 \beta q^{20} - q^{22} - 6 q^{23} + 13 q^{25} - 2 q^{29} + \beta q^{31} - q^{32} + 4 \beta q^{34} - 10 q^{37} - 3 \beta q^{40} - 8 \beta q^{41} - 8 q^{43} + q^{44} + 6 q^{46} + 3 \beta q^{47} - 13 q^{50} - 8 q^{53} + 3 \beta q^{55} + 2 q^{58} + \beta q^{59} + 2 \beta q^{61} - \beta q^{62} + q^{64} + 2 q^{67} - 4 \beta q^{68} + 2 q^{71} - 6 \beta q^{73} + 10 q^{74} + 16 q^{79} + 3 \beta q^{80} + 8 \beta q^{82} - 12 \beta q^{83} - 24 q^{85} + 8 q^{86} - q^{88} - 5 \beta q^{89} - 6 q^{92} - 3 \beta q^{94} + 7 \beta q^{97} +O(q^{100})$$ q - q^2 + q^4 + 3*b * q^5 - q^8 - 3*b * q^10 + q^11 + q^16 - 4*b * q^17 + 3*b * q^20 - q^22 - 6 * q^23 + 13 * q^25 - 2 * q^29 + b * q^31 - q^32 + 4*b * q^34 - 10 * q^37 - 3*b * q^40 - 8*b * q^41 - 8 * q^43 + q^44 + 6 * q^46 + 3*b * q^47 - 13 * q^50 - 8 * q^53 + 3*b * q^55 + 2 * q^58 + b * q^59 + 2*b * q^61 - b * q^62 + q^64 + 2 * q^67 - 4*b * q^68 + 2 * q^71 - 6*b * q^73 + 10 * q^74 + 16 * q^79 + 3*b * q^80 + 8*b * q^82 - 12*b * q^83 - 24 * q^85 + 8 * q^86 - q^88 - 5*b * q^89 - 6 * q^92 - 3*b * q^94 + 7*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{11} + 2 q^{16} - 2 q^{22} - 12 q^{23} + 26 q^{25} - 4 q^{29} - 2 q^{32} - 20 q^{37} - 16 q^{43} + 2 q^{44} + 12 q^{46} - 26 q^{50} - 16 q^{53} + 4 q^{58} + 2 q^{64} + 4 q^{67} + 4 q^{71} + 20 q^{74} + 32 q^{79} - 48 q^{85} + 16 q^{86} - 2 q^{88} - 12 q^{92}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 2 * q^11 + 2 * q^16 - 2 * q^22 - 12 * q^23 + 26 * q^25 - 4 * q^29 - 2 * q^32 - 20 * q^37 - 16 * q^43 + 2 * q^44 + 12 * q^46 - 26 * q^50 - 16 * q^53 + 4 * q^58 + 2 * q^64 + 4 * q^67 + 4 * q^71 + 20 * q^74 + 32 * q^79 - 48 * q^85 + 16 * q^86 - 2 * q^88 - 12 * q^92

## 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
 −1.41421 1.41421
−1.00000 0 1.00000 −4.24264 0 0 −1.00000 0 4.24264
1.2 −1.00000 0 1.00000 4.24264 0 0 −1.00000 0 −4.24264
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9702.2.a.cp 2
3.b odd 2 1 1078.2.a.u 2
7.b odd 2 1 inner 9702.2.a.cp 2
12.b even 2 1 8624.2.a.bs 2
21.c even 2 1 1078.2.a.u 2
21.g even 6 2 1078.2.e.p 4
21.h odd 6 2 1078.2.e.p 4
84.h odd 2 1 8624.2.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.u 2 3.b odd 2 1
1078.2.a.u 2 21.c even 2 1
1078.2.e.p 4 21.g even 6 2
1078.2.e.p 4 21.h odd 6 2
8624.2.a.bs 2 12.b even 2 1
8624.2.a.bs 2 84.h odd 2 1
9702.2.a.cp 2 1.a even 1 1 trivial
9702.2.a.cp 2 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9702))$$:

 $$T_{5}^{2} - 18$$ T5^2 - 18 $$T_{13}$$ T13 $$T_{17}^{2} - 32$$ T17^2 - 32 $$T_{19}$$ T19 $$T_{23} + 6$$ T23 + 6 $$T_{29} + 2$$ T29 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 18$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 32$$
$19$ $$T^{2}$$
$23$ $$(T + 6)^{2}$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2} - 2$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2} - 128$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2} - 18$$
$53$ $$(T + 8)^{2}$$
$59$ $$T^{2} - 2$$
$61$ $$T^{2} - 8$$
$67$ $$(T - 2)^{2}$$
$71$ $$(T - 2)^{2}$$
$73$ $$T^{2} - 72$$
$79$ $$(T - 16)^{2}$$
$83$ $$T^{2} - 288$$
$89$ $$T^{2} - 50$$
$97$ $$T^{2} - 98$$