# Properties

 Label 9680.2.a.by 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: 3.3.788.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 7x - 3$$ x^3 - x^2 - 7*x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} - q^{5} + (\beta_{2} - \beta_1 - 1) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q - b1 * q^3 - q^5 + (b2 - b1 - 1) * q^7 + (b2 + 2*b1 + 1) * q^9 $$q - \beta_1 q^{3} - q^{5} + (\beta_{2} - \beta_1 - 1) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9} + (\beta_{2} - 1) q^{13} + \beta_1 q^{15} + (2 \beta_{2} + 2 \beta_1) q^{17} + (\beta_{2} + 1) q^{19} + (2 \beta_{2} + 2 \beta_1 + 5) q^{21} + (\beta_{2} + 2 \beta_1 + 1) q^{23} + q^{25} + ( - \beta_{2} - 3 \beta_1 - 7) q^{27} + 2 \beta_1 q^{29} + (\beta_{2} + 7) q^{31} + ( - \beta_{2} + \beta_1 + 1) q^{35} + (2 \beta_1 - 2) q^{37} + (\beta_{2} + 1) q^{39} + ( - \beta_{2} + 2) q^{41} + (3 \beta_{2} + \beta_1 - 1) q^{43} + ( - \beta_{2} - 2 \beta_1 - 1) q^{45} + (2 \beta_{2} - \beta_1 + 2) q^{47} + (\beta_{2} + 7) q^{49} + ( - 6 \beta_1 - 6) q^{51} + ( - 3 \beta_{2} - 1) q^{53} + (\beta_{2} - 2 \beta_1 + 1) q^{57} + (\beta_{2} + 2 \beta_1 + 1) q^{59} + (\beta_{2} + 2 \beta_1 - 8) q^{61} + ( - 3 \beta_{2} - 8 \beta_1 - 3) q^{63} + ( - \beta_{2} + 1) q^{65} + (3 \beta_1 - 6) q^{67} + ( - \beta_{2} - 6 \beta_1 - 7) q^{69} + (5 \beta_{2} + 2 \beta_1 + 1) q^{71} + (2 \beta_{2} - 2) q^{73} - \beta_1 q^{75} + (2 \beta_{2} + 2 \beta_1 - 2) q^{79} + ( - \beta_{2} + 8 \beta_1 + 8) q^{81} + (\beta_{2} - 2 \beta_1 - 3) q^{83} + ( - 2 \beta_{2} - 2 \beta_1) q^{85} + ( - 2 \beta_{2} - 4 \beta_1 - 8) q^{87} + (2 \beta_{2} + 2 \beta_1 + 5) q^{89} + ( - \beta_{2} - 2 \beta_1 + 9) q^{91} + (\beta_{2} - 8 \beta_1 + 1) q^{93} + ( - \beta_{2} - 1) q^{95} + (\beta_{2} - 1) q^{97}+O(q^{100})$$ q - b1 * q^3 - q^5 + (b2 - b1 - 1) * q^7 + (b2 + 2*b1 + 1) * q^9 + (b2 - 1) * q^13 + b1 * q^15 + (2*b2 + 2*b1) * q^17 + (b2 + 1) * q^19 + (2*b2 + 2*b1 + 5) * q^21 + (b2 + 2*b1 + 1) * q^23 + q^25 + (-b2 - 3*b1 - 7) * q^27 + 2*b1 * q^29 + (b2 + 7) * q^31 + (-b2 + b1 + 1) * q^35 + (2*b1 - 2) * q^37 + (b2 + 1) * q^39 + (-b2 + 2) * q^41 + (3*b2 + b1 - 1) * q^43 + (-b2 - 2*b1 - 1) * q^45 + (2*b2 - b1 + 2) * q^47 + (b2 + 7) * q^49 + (-6*b1 - 6) * q^51 + (-3*b2 - 1) * q^53 + (b2 - 2*b1 + 1) * q^57 + (b2 + 2*b1 + 1) * q^59 + (b2 + 2*b1 - 8) * q^61 + (-3*b2 - 8*b1 - 3) * q^63 + (-b2 + 1) * q^65 + (3*b1 - 6) * q^67 + (-b2 - 6*b1 - 7) * q^69 + (5*b2 + 2*b1 + 1) * q^71 + (2*b2 - 2) * q^73 - b1 * q^75 + (2*b2 + 2*b1 - 2) * q^79 + (-b2 + 8*b1 + 8) * q^81 + (b2 - 2*b1 - 3) * q^83 + (-2*b2 - 2*b1) * q^85 + (-2*b2 - 4*b1 - 8) * q^87 + (2*b2 + 2*b1 + 5) * q^89 + (-b2 - 2*b1 + 9) * q^91 + (b2 - 8*b1 + 1) * q^93 + (-b2 - 1) * q^95 + (b2 - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10})$$ 3 * q - q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9 $$3 q - q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} - 2 q^{13} + q^{15} + 4 q^{17} + 4 q^{19} + 19 q^{21} + 6 q^{23} + 3 q^{25} - 25 q^{27} + 2 q^{29} + 22 q^{31} + 3 q^{35} - 4 q^{37} + 4 q^{39} + 5 q^{41} + q^{43} - 6 q^{45} + 7 q^{47} + 22 q^{49} - 24 q^{51} - 6 q^{53} + 2 q^{57} + 6 q^{59} - 21 q^{61} - 20 q^{63} + 2 q^{65} - 15 q^{67} - 28 q^{69} + 10 q^{71} - 4 q^{73} - q^{75} - 2 q^{79} + 31 q^{81} - 10 q^{83} - 4 q^{85} - 30 q^{87} + 19 q^{89} + 24 q^{91} - 4 q^{93} - 4 q^{95} - 2 q^{97}+O(q^{100})$$ 3 * q - q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9 - 2 * q^13 + q^15 + 4 * q^17 + 4 * q^19 + 19 * q^21 + 6 * q^23 + 3 * q^25 - 25 * q^27 + 2 * q^29 + 22 * q^31 + 3 * q^35 - 4 * q^37 + 4 * q^39 + 5 * q^41 + q^43 - 6 * q^45 + 7 * q^47 + 22 * q^49 - 24 * q^51 - 6 * q^53 + 2 * q^57 + 6 * q^59 - 21 * q^61 - 20 * q^63 + 2 * q^65 - 15 * q^67 - 28 * q^69 + 10 * q^71 - 4 * q^73 - q^75 - 2 * q^79 + 31 * q^81 - 10 * q^83 - 4 * q^85 - 30 * q^87 + 19 * q^89 + 24 * q^91 - 4 * q^93 - 4 * q^95 - 2 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7x - 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 4$$ v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 4$$ b2 + 2*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
 3.35386 −0.476452 −1.87740
0 −3.35386 0 −1.00000 0 −3.81322 0 8.24835 0
1.2 0 0.476452 0 −1.00000 0 −3.34364 0 −2.77299 0
1.3 0 1.87740 0 −1.00000 0 4.15686 0 0.524645 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.by 3
4.b odd 2 1 4840.2.a.v yes 3
11.b odd 2 1 9680.2.a.cd 3
44.c even 2 1 4840.2.a.s 3

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

 $$T_{3}^{3} + T_{3}^{2} - 7T_{3} + 3$$ T3^3 + T3^2 - 7*T3 + 3 $$T_{7}^{3} + 3T_{7}^{2} - 17T_{7} - 53$$ T7^3 + 3*T7^2 - 17*T7 - 53 $$T_{13}^{3} + 2T_{13}^{2} - 8T_{13} - 4$$ T13^3 + 2*T13^2 - 8*T13 - 4 $$T_{17}^{3} - 4T_{17}^{2} - 48T_{17} + 144$$ T17^3 - 4*T17^2 - 48*T17 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + T^{2} - 7T + 3$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + 3 T^{2} + \cdots - 53$$
$11$ $$T^{3}$$
$13$ $$T^{3} + 2 T^{2} + \cdots - 4$$
$17$ $$T^{3} - 4 T^{2} + \cdots + 144$$
$19$ $$T^{3} - 4 T^{2} + \cdots + 12$$
$23$ $$T^{3} - 6 T^{2} + \cdots + 12$$
$29$ $$T^{3} - 2 T^{2} + \cdots - 24$$
$31$ $$T^{3} - 22 T^{2} + \cdots - 324$$
$37$ $$T^{3} + 4 T^{2} + \cdots - 80$$
$41$ $$T^{3} - 5T^{2} - T + 9$$
$43$ $$T^{3} - T^{2} + \cdots + 275$$
$47$ $$T^{3} - 7 T^{2} + \cdots - 9$$
$53$ $$T^{3} + 6 T^{2} + \cdots - 212$$
$59$ $$T^{3} - 6 T^{2} + \cdots + 12$$
$61$ $$T^{3} + 21 T^{2} + \cdots + 75$$
$67$ $$T^{3} + 15 T^{2} + \cdots - 351$$
$71$ $$T^{3} - 10 T^{2} + \cdots + 1996$$
$73$ $$T^{3} + 4 T^{2} + \cdots - 32$$
$79$ $$T^{3} + 2 T^{2} + \cdots + 40$$
$83$ $$T^{3} + 10 T^{2} + \cdots - 180$$
$89$ $$T^{3} - 19 T^{2} + \cdots + 159$$
$97$ $$T^{3} + 2 T^{2} + \cdots - 4$$