# Properties

 Label 9680.2.a.cu Level $9680$ Weight $2$ Character orbit 9680.a Self dual yes Analytic conductor $77.295$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.725.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 3x^{2} + x + 1$$ x^4 - x^3 - 3*x^2 + x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + q^{5} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{7} + ( - \beta_{3} + \beta_{2} - 1) q^{9}+O(q^{10})$$ q + b2 * q^3 + q^5 + (b3 - b2 - b1 + 2) * q^7 + (-b3 + b2 - 1) * q^9 $$q + \beta_{2} q^{3} + q^{5} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{7} + ( - \beta_{3} + \beta_{2} - 1) q^{9} + (2 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{13} + \beta_{2} q^{15} + ( - 2 \beta_{3} + 3 \beta_1 + 3) q^{17} + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots - 1) q^{19}+ \cdots + (2 \beta_{3} - 6 \beta_1 + 1) q^{97}+O(q^{100})$$ q + b2 * q^3 + q^5 + (b3 - b2 - b1 + 2) * q^7 + (-b3 + b2 - 1) * q^9 + (2*b3 - b2 - 3*b1 + 1) * q^13 + b2 * q^15 + (-2*b3 + 3*b1 + 3) * q^17 + (-3*b3 + 3*b2 + 2*b1 - 1) * q^19 + (b2 - 1) * q^21 + (-b3 + b1 + 3) * q^23 + q^25 + (-b3 - 3*b2 - b1 + 2) * q^27 + (-3*b3 + 3*b2 + 1) * q^29 + (-2*b3 - 3*b2 - b1 + 7) * q^31 + (b3 - b2 - b1 + 2) * q^35 + (4*b3 + 4*b2 + b1 - 5) * q^37 + (-2*b3 - b1 + 1) * q^39 + (-2*b3 + 5*b1 + 3) * q^41 + (-2*b3 + 3*b2 + b1 + 1) * q^43 + (-b3 + b2 - 1) * q^45 + (b3 - 2*b2 + 3*b1) * q^47 + (6*b3 - 4*b2 - 5*b1 - 1) * q^49 + (3*b3 + 3*b2 + b1 - 3) * q^51 + (-6*b3 - 2*b2 + b1) * q^53 + (-b3 + 2*b2 - b1 + 4) * q^57 + (b3 - b2 - 7*b1 + 6) * q^59 + (7*b3 - 6*b1 - 1) * q^61 + (-4*b3 + 3*b2 + 3*b1 - 4) * q^63 + (2*b3 - b2 - 3*b1 + 1) * q^65 + (6*b3 + b2 - 5*b1 - 4) * q^67 + (b3 + 3*b2 - 1) * q^69 + (4*b3 - 3*b2 + 5*b1 + 2) * q^71 + (6*b3 + b2 - 7*b1) * q^73 + b2 * q^75 + (7*b3 + b2 - 1) * q^79 + (5*b3 - 4*b2 - 2*b1 - 2) * q^81 + (-b3 + b1 - 2) * q^83 + (-2*b3 + 3*b1 + 3) * q^85 + (-3*b3 + 4*b2 - 3*b1 + 6) * q^87 + (-2*b3 - 6*b2 + 1) * q^89 + (10*b3 - 4*b2 - 8*b1 + 5) * q^91 + (2*b3 + 4*b2 - 3*b1 - 5) * q^93 + (-3*b3 + 3*b2 + 2*b1 - 1) * q^95 + (2*b3 - 6*b1 + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 4 q^{5} + 7 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 4 * q^5 + 7 * q^7 - 4 * q^9 $$4 q + 2 q^{3} + 4 q^{5} + 7 q^{7} - 4 q^{9} + 3 q^{13} + 2 q^{15} + 11 q^{17} - 2 q^{19} - 2 q^{21} + 11 q^{23} + 4 q^{25} - q^{27} + 4 q^{29} + 17 q^{31} + 7 q^{35} - 3 q^{37} - q^{39} + 13 q^{41} + 7 q^{43} - 4 q^{45} + q^{47} - 5 q^{49} + q^{51} - 15 q^{53} + 17 q^{57} + 17 q^{59} + 4 q^{61} - 15 q^{63} + 3 q^{65} - 7 q^{67} + 4 q^{69} + 15 q^{71} + 7 q^{73} + 2 q^{75} + 12 q^{79} - 8 q^{81} - 9 q^{83} + 11 q^{85} + 23 q^{87} - 12 q^{89} + 24 q^{91} - 11 q^{93} - 2 q^{95} + 2 q^{97}+O(q^{100})$$ 4 * q + 2 * q^3 + 4 * q^5 + 7 * q^7 - 4 * q^9 + 3 * q^13 + 2 * q^15 + 11 * q^17 - 2 * q^19 - 2 * q^21 + 11 * q^23 + 4 * q^25 - q^27 + 4 * q^29 + 17 * q^31 + 7 * q^35 - 3 * q^37 - q^39 + 13 * q^41 + 7 * q^43 - 4 * q^45 + q^47 - 5 * q^49 + q^51 - 15 * q^53 + 17 * q^57 + 17 * q^59 + 4 * q^61 - 15 * q^63 + 3 * q^65 - 7 * q^67 + 4 * q^69 + 15 * q^71 + 7 * q^73 + 2 * q^75 + 12 * q^79 - 8 * q^81 - 9 * q^83 + 11 * q^85 + 23 * q^87 - 12 * q^89 + 24 * q^91 - 11 * q^93 - 2 * q^95 + 2 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 3x^{2} + x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 1$$ v^2 - v - 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 2\nu + 1$$ v^3 - v^2 - 2*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 1$$ b2 + b1 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 3\beta_1$$ b3 + b2 + 3*b1

## 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
 0.737640 −0.477260 2.09529 −1.35567
0 −1.19353 0 1.00000 0 1.83785 0 −1.57549 0
1.2 0 −0.294963 0 1.00000 0 4.39026 0 −2.91300 0
1.3 0 1.29496 0 1.00000 0 0.227777 0 −1.32307 0
1.4 0 2.19353 0 1.00000 0 0.544113 0 1.81156 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.cu 4
4.b odd 2 1 4840.2.a.y 4
11.b odd 2 1 9680.2.a.ct 4
11.c even 5 2 880.2.bo.d 8
44.c even 2 1 4840.2.a.z 4
44.h odd 10 2 440.2.y.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.y.a 8 44.h odd 10 2
880.2.bo.d 8 11.c even 5 2
4840.2.a.y 4 4.b odd 2 1
4840.2.a.z 4 44.c even 2 1
9680.2.a.ct 4 11.b odd 2 1
9680.2.a.cu 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9680))$$:

 $$T_{3}^{4} - 2T_{3}^{3} - 2T_{3}^{2} + 3T_{3} + 1$$ T3^4 - 2*T3^3 - 2*T3^2 + 3*T3 + 1 $$T_{7}^{4} - 7T_{7}^{3} + 13T_{7}^{2} - 7T_{7} + 1$$ T7^4 - 7*T7^3 + 13*T7^2 - 7*T7 + 1 $$T_{13}^{4} - 3T_{13}^{3} - 21T_{13}^{2} + 13T_{13} + 41$$ T13^4 - 3*T13^3 - 21*T13^2 + 13*T13 + 41 $$T_{17}^{4} - 11T_{17}^{3} + 20T_{17}^{2} + 62T_{17} - 11$$ T17^4 - 11*T17^3 + 20*T17^2 + 62*T17 - 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 2 T^{3} + \cdots + 1$$
$5$ $$(T - 1)^{4}$$
$7$ $$T^{4} - 7 T^{3} + \cdots + 1$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 3 T^{3} + \cdots + 41$$
$17$ $$T^{4} - 11 T^{3} + \cdots - 11$$
$19$ $$T^{4} + 2 T^{3} + \cdots + 101$$
$23$ $$T^{4} - 11 T^{3} + \cdots + 31$$
$29$ $$T^{4} - 4 T^{3} + \cdots + 1$$
$31$ $$T^{4} - 17 T^{3} + \cdots - 379$$
$37$ $$T^{4} + 3 T^{3} + \cdots - 1$$
$41$ $$T^{4} - 13 T^{3} + \cdots + 541$$
$43$ $$T^{4} - 7 T^{3} + \cdots + 61$$
$47$ $$T^{4} - T^{3} + \cdots - 149$$
$53$ $$T^{4} + 15 T^{3} + \cdots - 1361$$
$59$ $$T^{4} - 17 T^{3} + \cdots - 1681$$
$61$ $$T^{4} - 4 T^{3} + \cdots + 811$$
$67$ $$T^{4} + 7 T^{3} + \cdots + 431$$
$71$ $$T^{4} - 15 T^{3} + \cdots - 9871$$
$73$ $$T^{4} - 7 T^{3} + \cdots + 3751$$
$79$ $$T^{4} - 12 T^{3} + \cdots + 2381$$
$83$ $$T^{4} + 9 T^{3} + \cdots + 11$$
$89$ $$T^{4} + 12 T^{3} + \cdots - 479$$
$97$ $$T^{4} - 2 T^{3} + \cdots + 2179$$