# Properties

 Label 9702.2.a.ch Level $9702$ Weight $2$ Character orbit 9702.a Self dual yes Analytic conductor $77.471$ Analytic rank $1$ Dimension $2$ 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: $$2$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ 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)

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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta - 2) q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (b - 2) * q^5 - q^8 $$q - q^{2} + q^{4} + (\beta - 2) q^{5} - q^{8} + ( - \beta + 2) q^{10} - q^{11} + (2 \beta + 1) q^{13} + q^{16} + (4 \beta - 2) q^{17} + ( - \beta + 2) q^{19} + (\beta - 2) q^{20} + q^{22} + (3 \beta + 2) q^{23} + ( - 4 \beta + 1) q^{25} + ( - 2 \beta - 1) q^{26} + ( - 4 \beta + 3) q^{29} + 4 q^{31} - q^{32} + ( - 4 \beta + 2) q^{34} + ( - \beta - 8) q^{37} + (\beta - 2) q^{38} + ( - \beta + 2) q^{40} + ( - \beta - 4) q^{41} - 4 \beta q^{43} - q^{44} + ( - 3 \beta - 2) q^{46} + ( - 6 \beta - 2) q^{47} + (4 \beta - 1) q^{50} + (2 \beta + 1) q^{52} + ( - 7 \beta + 2) q^{53} + ( - \beta + 2) q^{55} + (4 \beta - 3) q^{58} + (\beta - 7) q^{59} + ( - 2 \beta - 9) q^{61} - 4 q^{62} + q^{64} + ( - 3 \beta + 2) q^{65} + ( - 3 \beta + 7) q^{67} + (4 \beta - 2) q^{68} + (5 \beta + 4) q^{71} + (\beta + 8) q^{73} + (\beta + 8) q^{74} + ( - \beta + 2) q^{76} + ( - 3 \beta - 9) q^{79} + (\beta - 2) q^{80} + (\beta + 4) q^{82} + ( - 10 \beta + 2) q^{83} + ( - 10 \beta + 12) q^{85} + 4 \beta q^{86} + q^{88} + (6 \beta + 4) q^{89} + (3 \beta + 2) q^{92} + (6 \beta + 2) q^{94} + (4 \beta - 6) q^{95} + (2 \beta + 1) q^{97} +O(q^{100})$$ q - q^2 + q^4 + (b - 2) * q^5 - q^8 + (-b + 2) * q^10 - q^11 + (2*b + 1) * q^13 + q^16 + (4*b - 2) * q^17 + (-b + 2) * q^19 + (b - 2) * q^20 + q^22 + (3*b + 2) * q^23 + (-4*b + 1) * q^25 + (-2*b - 1) * q^26 + (-4*b + 3) * q^29 + 4 * q^31 - q^32 + (-4*b + 2) * q^34 + (-b - 8) * q^37 + (b - 2) * q^38 + (-b + 2) * q^40 + (-b - 4) * q^41 - 4*b * q^43 - q^44 + (-3*b - 2) * q^46 + (-6*b - 2) * q^47 + (4*b - 1) * q^50 + (2*b + 1) * q^52 + (-7*b + 2) * q^53 + (-b + 2) * q^55 + (4*b - 3) * q^58 + (b - 7) * q^59 + (-2*b - 9) * q^61 - 4 * q^62 + q^64 + (-3*b + 2) * q^65 + (-3*b + 7) * q^67 + (4*b - 2) * q^68 + (5*b + 4) * q^71 + (b + 8) * q^73 + (b + 8) * q^74 + (-b + 2) * q^76 + (-3*b - 9) * q^79 + (b - 2) * q^80 + (b + 4) * q^82 + (-10*b + 2) * q^83 + (-10*b + 12) * q^85 + 4*b * q^86 + q^88 + (6*b + 4) * q^89 + (3*b + 2) * q^92 + (6*b + 2) * q^94 + (4*b - 6) * q^95 + (2*b + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 4 q^{10} - 2 q^{11} + 2 q^{13} + 2 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} + 2 q^{22} + 4 q^{23} + 2 q^{25} - 2 q^{26} + 6 q^{29} + 8 q^{31} - 2 q^{32} + 4 q^{34} - 16 q^{37} - 4 q^{38} + 4 q^{40} - 8 q^{41} - 2 q^{44} - 4 q^{46} - 4 q^{47} - 2 q^{50} + 2 q^{52} + 4 q^{53} + 4 q^{55} - 6 q^{58} - 14 q^{59} - 18 q^{61} - 8 q^{62} + 2 q^{64} + 4 q^{65} + 14 q^{67} - 4 q^{68} + 8 q^{71} + 16 q^{73} + 16 q^{74} + 4 q^{76} - 18 q^{79} - 4 q^{80} + 8 q^{82} + 4 q^{83} + 24 q^{85} + 2 q^{88} + 8 q^{89} + 4 q^{92} + 4 q^{94} - 12 q^{95} + 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 + 4 * q^10 - 2 * q^11 + 2 * q^13 + 2 * q^16 - 4 * q^17 + 4 * q^19 - 4 * q^20 + 2 * q^22 + 4 * q^23 + 2 * q^25 - 2 * q^26 + 6 * q^29 + 8 * q^31 - 2 * q^32 + 4 * q^34 - 16 * q^37 - 4 * q^38 + 4 * q^40 - 8 * q^41 - 2 * q^44 - 4 * q^46 - 4 * q^47 - 2 * q^50 + 2 * q^52 + 4 * q^53 + 4 * q^55 - 6 * q^58 - 14 * q^59 - 18 * q^61 - 8 * q^62 + 2 * q^64 + 4 * q^65 + 14 * q^67 - 4 * q^68 + 8 * q^71 + 16 * q^73 + 16 * q^74 + 4 * q^76 - 18 * q^79 - 4 * q^80 + 8 * q^82 + 4 * q^83 + 24 * q^85 + 2 * q^88 + 8 * q^89 + 4 * q^92 + 4 * q^94 - 12 * q^95 + 2 * 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
 −1.41421 1.41421
−1.00000 0 1.00000 −3.41421 0 0 −1.00000 0 3.41421
1.2 −1.00000 0 1.00000 −0.585786 0 0 −1.00000 0 0.585786
 $$n$$: 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.ch 2
3.b odd 2 1 1078.2.a.x 2
7.b odd 2 1 9702.2.a.cx 2
7.d odd 6 2 1386.2.k.t 4
12.b even 2 1 8624.2.a.bh 2
21.c even 2 1 1078.2.a.t 2
21.g even 6 2 154.2.e.e 4
21.h odd 6 2 1078.2.e.m 4
84.h odd 2 1 8624.2.a.cc 2
84.j odd 6 2 1232.2.q.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.e 4 21.g even 6 2
1078.2.a.t 2 21.c even 2 1
1078.2.a.x 2 3.b odd 2 1
1078.2.e.m 4 21.h odd 6 2
1232.2.q.f 4 84.j odd 6 2
1386.2.k.t 4 7.d odd 6 2
8624.2.a.bh 2 12.b even 2 1
8624.2.a.cc 2 84.h odd 2 1
9702.2.a.ch 2 1.a even 1 1 trivial
9702.2.a.cx 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9702))$$:

 $$T_{5}^{2} + 4T_{5} + 2$$ T5^2 + 4*T5 + 2 $$T_{13}^{2} - 2T_{13} - 7$$ T13^2 - 2*T13 - 7 $$T_{17}^{2} + 4T_{17} - 28$$ T17^2 + 4*T17 - 28 $$T_{19}^{2} - 4T_{19} + 2$$ T19^2 - 4*T19 + 2 $$T_{23}^{2} - 4T_{23} - 14$$ T23^2 - 4*T23 - 14 $$T_{29}^{2} - 6T_{29} - 23$$ T29^2 - 6*T29 - 23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4T + 2$$
$7$ $$T^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 2T - 7$$
$17$ $$T^{2} + 4T - 28$$
$19$ $$T^{2} - 4T + 2$$
$23$ $$T^{2} - 4T - 14$$
$29$ $$T^{2} - 6T - 23$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 16T + 62$$
$41$ $$T^{2} + 8T + 14$$
$43$ $$T^{2} - 32$$
$47$ $$T^{2} + 4T - 68$$
$53$ $$T^{2} - 4T - 94$$
$59$ $$T^{2} + 14T + 47$$
$61$ $$T^{2} + 18T + 73$$
$67$ $$T^{2} - 14T + 31$$
$71$ $$T^{2} - 8T - 34$$
$73$ $$T^{2} - 16T + 62$$
$79$ $$T^{2} + 18T + 63$$
$83$ $$T^{2} - 4T - 196$$
$89$ $$T^{2} - 8T - 56$$
$97$ $$T^{2} - 2T - 7$$