# Properties

 Label 9680.2.a.z Level $9680$ Weight $2$ Character orbit 9680.a Self dual yes Analytic conductor $77.295$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} - q^{5} + q^{9}+O(q^{10})$$ q + 2 * q^3 - q^5 + q^9 $$q + 2 q^{3} - q^{5} + q^{9} + 2 q^{13} - 2 q^{15} - 6 q^{17} + 2 q^{23} + q^{25} - 4 q^{27} + 8 q^{29} - 4 q^{31} - 2 q^{37} + 4 q^{39} + 4 q^{41} - 4 q^{43} - q^{45} - 2 q^{47} - 7 q^{49} - 12 q^{51} - 10 q^{53} - 8 q^{61} - 2 q^{65} + 2 q^{67} + 4 q^{69} - 8 q^{71} + 10 q^{73} + 2 q^{75} - 4 q^{79} - 11 q^{81} + 12 q^{83} + 6 q^{85} + 16 q^{87} + 6 q^{89} - 8 q^{93} + 6 q^{97}+O(q^{100})$$ q + 2 * q^3 - q^5 + q^9 + 2 * q^13 - 2 * q^15 - 6 * q^17 + 2 * q^23 + q^25 - 4 * q^27 + 8 * q^29 - 4 * q^31 - 2 * q^37 + 4 * q^39 + 4 * q^41 - 4 * q^43 - q^45 - 2 * q^47 - 7 * q^49 - 12 * q^51 - 10 * q^53 - 8 * q^61 - 2 * q^65 + 2 * q^67 + 4 * q^69 - 8 * q^71 + 10 * q^73 + 2 * q^75 - 4 * q^79 - 11 * q^81 + 12 * q^83 + 6 * q^85 + 16 * q^87 + 6 * q^89 - 8 * q^93 + 6 * 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
 0
0 2.00000 0 −1.00000 0 0 0 1.00000 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.z 1
4.b odd 2 1 4840.2.a.b yes 1
11.b odd 2 1 9680.2.a.y 1
44.c even 2 1 4840.2.a.a 1

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

 $$T_{3} - 2$$ T3 - 2 $$T_{7}$$ T7 $$T_{13} - 2$$ T13 - 2 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 2$$
$17$ $$T + 6$$
$19$ $$T$$
$23$ $$T - 2$$
$29$ $$T - 8$$
$31$ $$T + 4$$
$37$ $$T + 2$$
$41$ $$T - 4$$
$43$ $$T + 4$$
$47$ $$T + 2$$
$53$ $$T + 10$$
$59$ $$T$$
$61$ $$T + 8$$
$67$ $$T - 2$$
$71$ $$T + 8$$
$73$ $$T - 10$$
$79$ $$T + 4$$
$83$ $$T - 12$$
$89$ $$T - 6$$
$97$ $$T - 6$$