# Properties

 Label 9680.2.a.ch Level $9680$ Weight $2$ Character orbit 9680.a Self dual yes Analytic conductor $77.295$ 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] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9680 = 2^{4} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9680.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: $$77.2951891566$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 4840) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + q^{5} - \beta_{2} q^{7} + ( - \beta_1 + 3) q^{9}+O(q^{10})$$ q - b2 * q^3 + q^5 - b2 * q^7 + (-b1 + 3) * q^9 $$q - \beta_{2} q^{3} + q^{5} - \beta_{2} q^{7} + ( - \beta_1 + 3) q^{9} + (\beta_{2} + \beta_1) q^{13} - \beta_{2} q^{15} + ( - \beta_{2} + 2) q^{17} + ( - 2 \beta_{2} + \beta_1 - 2) q^{19} + ( - \beta_1 + 6) q^{21} + (\beta_{2} - \beta_1 + 2) q^{23} + q^{25} + ( - 2 \beta_{2} - \beta_1 + 2) q^{27} + ( - \beta_1 + 2) q^{29} + ( - \beta_1 + 6) q^{31} - \beta_{2} q^{35} + \beta_1 q^{37} + (2 \beta_{2} + 2 \beta_1 - 8) q^{39} + 2 \beta_{2} q^{41} + ( - \beta_{2} + \beta_1 - 6) q^{43} + ( - \beta_1 + 3) q^{45} + ( - \beta_{2} - \beta_1 + 2) q^{47} + ( - \beta_1 - 1) q^{49} + ( - 2 \beta_{2} - \beta_1 + 6) q^{51} + \beta_1 q^{53} + (4 \beta_{2} - \beta_1 + 10) q^{57} + ( - 2 \beta_{2} + 2 \beta_1) q^{59} + ( - 4 \beta_{2} + \beta_1 + 2) q^{61} + ( - 5 \beta_{2} - \beta_1 + 2) q^{63} + (\beta_{2} + \beta_1) q^{65} + (3 \beta_{2} + \beta_1 + 6) q^{67} + ( - 4 \beta_{2} - 4) q^{69} + ( - 4 \beta_{2} + \beta_1 - 2) q^{71} + ( - \beta_{2} + \beta_1 - 8) q^{73} - \beta_{2} q^{75} - 12 q^{79} + ( - 4 \beta_{2} + 5) q^{81} + (3 \beta_{2} - 3 \beta_1 + 2) q^{83} + ( - \beta_{2} + 2) q^{85} + ( - 4 \beta_{2} - \beta_1 + 2) q^{87} + (4 \beta_{2} + \beta_1) q^{89} + (2 \beta_{2} + 2 \beta_1 - 8) q^{91} + ( - 8 \beta_{2} - \beta_1 + 2) q^{93} + ( - 2 \beta_{2} + \beta_1 - 2) q^{95} + 6 q^{97}+O(q^{100})$$ q - b2 * q^3 + q^5 - b2 * q^7 + (-b1 + 3) * q^9 + (b2 + b1) * q^13 - b2 * q^15 + (-b2 + 2) * q^17 + (-2*b2 + b1 - 2) * q^19 + (-b1 + 6) * q^21 + (b2 - b1 + 2) * q^23 + q^25 + (-2*b2 - b1 + 2) * q^27 + (-b1 + 2) * q^29 + (-b1 + 6) * q^31 - b2 * q^35 + b1 * q^37 + (2*b2 + 2*b1 - 8) * q^39 + 2*b2 * q^41 + (-b2 + b1 - 6) * q^43 + (-b1 + 3) * q^45 + (-b2 - b1 + 2) * q^47 + (-b1 - 1) * q^49 + (-2*b2 - b1 + 6) * q^51 + b1 * q^53 + (4*b2 - b1 + 10) * q^57 + (-2*b2 + 2*b1) * q^59 + (-4*b2 + b1 + 2) * q^61 + (-5*b2 - b1 + 2) * q^63 + (b2 + b1) * q^65 + (3*b2 + b1 + 6) * q^67 + (-4*b2 - 4) * q^69 + (-4*b2 + b1 - 2) * q^71 + (-b2 + b1 - 8) * q^73 - b2 * q^75 - 12 * q^79 + (-4*b2 + 5) * q^81 + (3*b2 - 3*b1 + 2) * q^83 + (-b2 + 2) * q^85 + (-4*b2 - b1 + 2) * q^87 + (4*b2 + b1) * q^89 + (2*b2 + 2*b1 - 8) * q^91 + (-8*b2 - b1 + 2) * q^93 + (-2*b2 + b1 - 2) * q^95 + 6 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} + 3 q^{5} + q^{7} + 8 q^{9}+O(q^{10})$$ 3 * q + q^3 + 3 * q^5 + q^7 + 8 * q^9 $$3 q + q^{3} + 3 q^{5} + q^{7} + 8 q^{9} + q^{15} + 7 q^{17} - 3 q^{19} + 17 q^{21} + 4 q^{23} + 3 q^{25} + 7 q^{27} + 5 q^{29} + 17 q^{31} + q^{35} + q^{37} - 24 q^{39} - 2 q^{41} - 16 q^{43} + 8 q^{45} + 6 q^{47} - 4 q^{49} + 19 q^{51} + q^{53} + 25 q^{57} + 4 q^{59} + 11 q^{61} + 10 q^{63} + 16 q^{67} - 8 q^{69} - q^{71} - 22 q^{73} + q^{75} - 36 q^{79} + 19 q^{81} + 7 q^{85} + 9 q^{87} - 3 q^{89} - 24 q^{91} + 13 q^{93} - 3 q^{95} + 18 q^{97}+O(q^{100})$$ 3 * q + q^3 + 3 * q^5 + q^7 + 8 * q^9 + q^15 + 7 * q^17 - 3 * q^19 + 17 * q^21 + 4 * q^23 + 3 * q^25 + 7 * q^27 + 5 * q^29 + 17 * q^31 + q^35 + q^37 - 24 * q^39 - 2 * q^41 - 16 * q^43 + 8 * q^45 + 6 * q^47 - 4 * q^49 + 19 * q^51 + q^53 + 25 * q^57 + 4 * q^59 + 11 * q^61 + 10 * q^63 + 16 * q^67 - 8 * q^69 - q^71 - 22 * q^73 + q^75 - 36 * q^79 + 19 * q^81 + 7 * q^85 + 9 * q^87 - 3 * q^89 - 24 * q^91 + 13 * q^93 - 3 * q^95 + 18 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 4$$ v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta_1 ) / 2$$ (-b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 4$$ b1 + 4

## 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
 −1.76156 3.12489 −0.363328
0 −2.62620 0 1.00000 0 −2.62620 0 3.89692 0
1.2 0 0.484862 0 1.00000 0 0.484862 0 −2.76491 0
1.3 0 3.14134 0 1.00000 0 3.14134 0 6.86799 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$5$$ $$-1$$
$$11$$ $$+1$$

## 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 9680.2.a.ch 3
4.b odd 2 1 4840.2.a.q 3
11.b odd 2 1 9680.2.a.cg 3
44.c even 2 1 4840.2.a.r yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4840.2.a.q 3 4.b odd 2 1
4840.2.a.r yes 3 44.c even 2 1
9680.2.a.cg 3 11.b odd 2 1
9680.2.a.ch 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9680))$$:

 $$T_{3}^{3} - T_{3}^{2} - 8T_{3} + 4$$ T3^3 - T3^2 - 8*T3 + 4 $$T_{7}^{3} - T_{7}^{2} - 8T_{7} + 4$$ T7^3 - T7^2 - 8*T7 + 4 $$T_{13}^{3} - 40T_{13} + 64$$ T13^3 - 40*T13 + 64 $$T_{17}^{3} - 7T_{17}^{2} + 8T_{17} + 8$$ T17^3 - 7*T17^2 + 8*T17 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - T^{2} - 8T + 4$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - T^{2} - 8T + 4$$
$11$ $$T^{3}$$
$13$ $$T^{3} - 40T + 64$$
$17$ $$T^{3} - 7 T^{2} + \cdots + 8$$
$19$ $$T^{3} + 3 T^{2} + \cdots + 16$$
$23$ $$T^{3} - 4 T^{2} + \cdots + 64$$
$29$ $$T^{3} - 5 T^{2} + \cdots + 64$$
$31$ $$T^{3} - 17 T^{2} + \cdots - 16$$
$37$ $$T^{3} - T^{2} + \cdots - 20$$
$41$ $$T^{3} + 2 T^{2} + \cdots - 32$$
$43$ $$T^{3} + 16 T^{2} + \cdots - 16$$
$47$ $$T^{3} - 6 T^{2} + \cdots + 8$$
$53$ $$T^{3} - T^{2} + \cdots - 20$$
$59$ $$T^{3} - 4 T^{2} + \cdots - 128$$
$61$ $$T^{3} - 11 T^{2} + \cdots + 976$$
$67$ $$T^{3} - 16 T^{2} + \cdots + 976$$
$71$ $$T^{3} + T^{2} + \cdots + 512$$
$73$ $$T^{3} + 22 T^{2} + \cdots + 176$$
$79$ $$(T + 12)^{3}$$
$83$ $$T^{3} - 228T + 880$$
$89$ $$T^{3} + 3 T^{2} + \cdots + 604$$
$97$ $$(T - 6)^{3}$$