# Properties

 Label 9702.2.a.dw Level $9702$ Weight $2$ Character orbit 9702.a Self dual yes Analytic conductor $77.471$ Analytic rank $0$ Dimension $3$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2700.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 15x - 20$$ x^3 - 15*x - 20 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + \beta_1 q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + b1 * q^5 - q^8 $$q - q^{2} + q^{4} + \beta_1 q^{5} - q^{8} - \beta_1 q^{10} + q^{11} + q^{16} + (2 \beta_{2} + 1) q^{17} + (\beta_{2} + 3) q^{19} + \beta_1 q^{20} - q^{22} + (\beta_{2} - \beta_1 - 1) q^{23} + (\beta_{2} + 2 \beta_1 + 5) q^{25} + (\beta_{2} + 2 \beta_1 - 3) q^{29} + (2 \beta_{2} + 2) q^{31} - q^{32} + ( - 2 \beta_{2} - 1) q^{34} + (\beta_{2} + 7) q^{37} + ( - \beta_{2} - 3) q^{38} - \beta_1 q^{40} + ( - \beta_{2} + 2 \beta_1 + 4) q^{41} + (3 \beta_{2} + 1) q^{43} + q^{44} + ( - \beta_{2} + \beta_1 + 1) q^{46} + ( - \beta_{2} - \beta_1 + 1) q^{47} + ( - \beta_{2} - 2 \beta_1 - 5) q^{50} - 8 q^{53} + \beta_1 q^{55} + ( - \beta_{2} - 2 \beta_1 + 3) q^{58} + ( - \beta_{2} - 2 \beta_1 - 3) q^{59} + ( - 2 \beta_{2} - \beta_1 + 2) q^{61} + ( - 2 \beta_{2} - 2) q^{62} + q^{64} + ( - \beta_{2} + 2 \beta_1 + 2) q^{67} + (2 \beta_{2} + 1) q^{68} + (3 \beta_{2} - 3) q^{71} + ( - 2 \beta_{2} + 2 \beta_1) q^{73} + ( - \beta_{2} - 7) q^{74} + (\beta_{2} + 3) q^{76} + ( - \beta_1 + 4) q^{79} + \beta_1 q^{80} + (\beta_{2} - 2 \beta_1 - 4) q^{82} + (\beta_{2} - 2 \beta_1 + 6) q^{83} + ( - 4 \beta_{2} + 3 \beta_1) q^{85} + ( - 3 \beta_{2} - 1) q^{86} - q^{88} + ( - 2 \beta_1 - 4) q^{89} + (\beta_{2} - \beta_1 - 1) q^{92} + (\beta_{2} + \beta_1 - 1) q^{94} + ( - 2 \beta_{2} + 4 \beta_1) q^{95} - 7 q^{97}+O(q^{100})$$ q - q^2 + q^4 + b1 * q^5 - q^8 - b1 * q^10 + q^11 + q^16 + (2*b2 + 1) * q^17 + (b2 + 3) * q^19 + b1 * q^20 - q^22 + (b2 - b1 - 1) * q^23 + (b2 + 2*b1 + 5) * q^25 + (b2 + 2*b1 - 3) * q^29 + (2*b2 + 2) * q^31 - q^32 + (-2*b2 - 1) * q^34 + (b2 + 7) * q^37 + (-b2 - 3) * q^38 - b1 * q^40 + (-b2 + 2*b1 + 4) * q^41 + (3*b2 + 1) * q^43 + q^44 + (-b2 + b1 + 1) * q^46 + (-b2 - b1 + 1) * q^47 + (-b2 - 2*b1 - 5) * q^50 - 8 * q^53 + b1 * q^55 + (-b2 - 2*b1 + 3) * q^58 + (-b2 - 2*b1 - 3) * q^59 + (-2*b2 - b1 + 2) * q^61 + (-2*b2 - 2) * q^62 + q^64 + (-b2 + 2*b1 + 2) * q^67 + (2*b2 + 1) * q^68 + (3*b2 - 3) * q^71 + (-2*b2 + 2*b1) * q^73 + (-b2 - 7) * q^74 + (b2 + 3) * q^76 + (-b1 + 4) * q^79 + b1 * q^80 + (b2 - 2*b1 - 4) * q^82 + (b2 - 2*b1 + 6) * q^83 + (-4*b2 + 3*b1) * q^85 + (-3*b2 - 1) * q^86 - q^88 + (-2*b1 - 4) * q^89 + (b2 - b1 - 1) * q^92 + (b2 + b1 - 1) * q^94 + (-2*b2 + 4*b1) * q^95 - 7 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 $$3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} + 3 q^{11} + 3 q^{16} + 3 q^{17} + 9 q^{19} - 3 q^{22} - 3 q^{23} + 15 q^{25} - 9 q^{29} + 6 q^{31} - 3 q^{32} - 3 q^{34} + 21 q^{37} - 9 q^{38} + 12 q^{41} + 3 q^{43} + 3 q^{44} + 3 q^{46} + 3 q^{47} - 15 q^{50} - 24 q^{53} + 9 q^{58} - 9 q^{59} + 6 q^{61} - 6 q^{62} + 3 q^{64} + 6 q^{67} + 3 q^{68} - 9 q^{71} - 21 q^{74} + 9 q^{76} + 12 q^{79} - 12 q^{82} + 18 q^{83} - 3 q^{86} - 3 q^{88} - 12 q^{89} - 3 q^{92} - 3 q^{94} - 21 q^{97}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 + 3 * q^11 + 3 * q^16 + 3 * q^17 + 9 * q^19 - 3 * q^22 - 3 * q^23 + 15 * q^25 - 9 * q^29 + 6 * q^31 - 3 * q^32 - 3 * q^34 + 21 * q^37 - 9 * q^38 + 12 * q^41 + 3 * q^43 + 3 * q^44 + 3 * q^46 + 3 * q^47 - 15 * q^50 - 24 * q^53 + 9 * q^58 - 9 * q^59 + 6 * q^61 - 6 * q^62 + 3 * q^64 + 6 * q^67 + 3 * q^68 - 9 * q^71 - 21 * q^74 + 9 * q^76 + 12 * q^79 - 12 * q^82 + 18 * q^83 - 3 * q^86 - 3 * q^88 - 12 * q^89 - 3 * q^92 - 3 * q^94 - 21 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 15x - 20$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 10$$ v^2 - 2*v - 10
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 10$$ b2 + 2*b1 + 10

## 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
 −2.80560 −1.61323 4.41883
−1.00000 0 1.00000 −2.80560 0 0 −1.00000 0 2.80560
1.2 −1.00000 0 1.00000 −1.61323 0 0 −1.00000 0 1.61323
1.3 −1.00000 0 1.00000 4.41883 0 0 −1.00000 0 −4.41883
 $$n$$: 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.dw 3
3.b odd 2 1 3234.2.a.bh 3
7.b odd 2 1 9702.2.a.dv 3
7.d odd 6 2 1386.2.k.v 6
21.c even 2 1 3234.2.a.bf 3
21.g even 6 2 462.2.i.g 6

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

 $$T_{5}^{3} - 15T_{5} - 20$$ T5^3 - 15*T5 - 20 $$T_{13}$$ T13 $$T_{17}^{3} - 3T_{17}^{2} - 57T_{17} + 139$$ T17^3 - 3*T17^2 - 57*T17 + 139 $$T_{19}^{3} - 9T_{19}^{2} + 12T_{19} + 28$$ T19^3 - 9*T19^2 + 12*T19 + 28 $$T_{23}^{3} + 3T_{23}^{2} - 27T_{23} - 89$$ T23^3 + 3*T23^2 - 27*T23 - 89 $$T_{29}^{3} + 9T_{29}^{2} - 48T_{29} - 348$$ T29^3 + 9*T29^2 - 48*T29 - 348

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3} - 15T - 20$$
$7$ $$T^{3}$$
$11$ $$(T - 1)^{3}$$
$13$ $$T^{3}$$
$17$ $$T^{3} - 3 T^{2} - 57 T + 139$$
$19$ $$T^{3} - 9 T^{2} + 12 T + 28$$
$23$ $$T^{3} + 3 T^{2} - 27 T - 89$$
$29$ $$T^{3} + 9 T^{2} - 48 T - 348$$
$31$ $$T^{3} - 6 T^{2} - 48 T + 192$$
$37$ $$T^{3} - 21 T^{2} + 132 T - 228$$
$41$ $$T^{3} - 12 T^{2} - 27 T + 306$$
$43$ $$T^{3} - 3 T^{2} - 132 T + 404$$
$47$ $$T^{3} - 3 T^{2} - 27 T + 9$$
$53$ $$(T + 8)^{3}$$
$59$ $$T^{3} + 9 T^{2} - 48 T - 48$$
$61$ $$T^{3} - 6 T^{2} - 63 T - 98$$
$67$ $$T^{3} - 6 T^{2} - 63 T + 212$$
$71$ $$T^{3} + 9 T^{2} - 108 T - 108$$
$73$ $$T^{3} - 120T + 480$$
$79$ $$T^{3} - 12 T^{2} + 33 T + 16$$
$83$ $$T^{3} - 18 T^{2} + 33 T + 164$$
$89$ $$T^{3} + 12 T^{2} - 12 T - 16$$
$97$ $$(T + 7)^{3}$$