# Properties

 Label 9680.2.a.bj Level $9680$ Weight $2$ Character orbit 9680.a Self dual yes Analytic conductor $77.295$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{3} + q^{5} + (\beta - 1) q^{7} - 2 \beta q^{9} +O(q^{10})$$ q + (b - 1) * q^3 + q^5 + (b - 1) * q^7 - 2*b * q^9 $$q + (\beta - 1) q^{3} + q^{5} + (\beta - 1) q^{7} - 2 \beta q^{9} + 2 q^{13} + (\beta - 1) q^{15} - 4 \beta q^{17} + (2 \beta - 2) q^{19} + ( - 2 \beta + 3) q^{21} + (4 \beta + 2) q^{23} + q^{25} + ( - \beta - 1) q^{27} + 2 q^{29} - 4 \beta q^{31} + (\beta - 1) q^{35} - 4 q^{37} + (2 \beta - 2) q^{39} + (6 \beta - 1) q^{41} + ( - \beta - 5) q^{43} - 2 \beta q^{45} + ( - \beta - 9) q^{47} + ( - 2 \beta - 4) q^{49} + (4 \beta - 8) q^{51} + (4 \beta - 2) q^{53} + ( - 4 \beta + 6) q^{57} - 8 \beta q^{59} + q^{61} + (2 \beta - 4) q^{63} + 2 q^{65} + ( - \beta + 5) q^{67} + ( - 2 \beta + 6) q^{69} + 4 q^{71} + 4 q^{73} + (\beta - 1) q^{75} + ( - 6 \beta + 6) q^{79} + (6 \beta - 1) q^{81} + ( - 8 \beta + 2) q^{83} - 4 \beta q^{85} + (2 \beta - 2) q^{87} + ( - 4 \beta - 3) q^{89} + (2 \beta - 2) q^{91} + (4 \beta - 8) q^{93} + (2 \beta - 2) q^{95} + (8 \beta - 6) q^{97} +O(q^{100})$$ q + (b - 1) * q^3 + q^5 + (b - 1) * q^7 - 2*b * q^9 + 2 * q^13 + (b - 1) * q^15 - 4*b * q^17 + (2*b - 2) * q^19 + (-2*b + 3) * q^21 + (4*b + 2) * q^23 + q^25 + (-b - 1) * q^27 + 2 * q^29 - 4*b * q^31 + (b - 1) * q^35 - 4 * q^37 + (2*b - 2) * q^39 + (6*b - 1) * q^41 + (-b - 5) * q^43 - 2*b * q^45 + (-b - 9) * q^47 + (-2*b - 4) * q^49 + (4*b - 8) * q^51 + (4*b - 2) * q^53 + (-4*b + 6) * q^57 - 8*b * q^59 + q^61 + (2*b - 4) * q^63 + 2 * q^65 + (-b + 5) * q^67 + (-2*b + 6) * q^69 + 4 * q^71 + 4 * q^73 + (b - 1) * q^75 + (-6*b + 6) * q^79 + (6*b - 1) * q^81 + (-8*b + 2) * q^83 - 4*b * q^85 + (2*b - 2) * q^87 + (-4*b - 3) * q^89 + (2*b - 2) * q^91 + (4*b - 8) * q^93 + (2*b - 2) * q^95 + (8*b - 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 $$2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{13} - 2 q^{15} - 4 q^{19} + 6 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{29} - 2 q^{35} - 8 q^{37} - 4 q^{39} - 2 q^{41} - 10 q^{43} - 18 q^{47} - 8 q^{49} - 16 q^{51} - 4 q^{53} + 12 q^{57} + 2 q^{61} - 8 q^{63} + 4 q^{65} + 10 q^{67} + 12 q^{69} + 8 q^{71} + 8 q^{73} - 2 q^{75} + 12 q^{79} - 2 q^{81} + 4 q^{83} - 4 q^{87} - 6 q^{89} - 4 q^{91} - 16 q^{93} - 4 q^{95} - 12 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^13 - 2 * q^15 - 4 * q^19 + 6 * q^21 + 4 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^29 - 2 * q^35 - 8 * q^37 - 4 * q^39 - 2 * q^41 - 10 * q^43 - 18 * q^47 - 8 * q^49 - 16 * q^51 - 4 * q^53 + 12 * q^57 + 2 * q^61 - 8 * q^63 + 4 * q^65 + 10 * q^67 + 12 * q^69 + 8 * q^71 + 8 * q^73 - 2 * q^75 + 12 * q^79 - 2 * q^81 + 4 * q^83 - 4 * q^87 - 6 * q^89 - 4 * q^91 - 16 * q^93 - 4 * q^95 - 12 * 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
 −1.41421 1.41421
0 −2.41421 0 1.00000 0 −2.41421 0 2.82843 0
1.2 0 0.414214 0 1.00000 0 0.414214 0 −2.82843 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.bj 2
4.b odd 2 1 4840.2.a.p yes 2
11.b odd 2 1 9680.2.a.bk 2
44.c even 2 1 4840.2.a.o 2

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

 $$T_{3}^{2} + 2T_{3} - 1$$ T3^2 + 2*T3 - 1 $$T_{7}^{2} + 2T_{7} - 1$$ T7^2 + 2*T7 - 1 $$T_{13} - 2$$ T13 - 2 $$T_{17}^{2} - 32$$ T17^2 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T - 1$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 2T - 1$$
$11$ $$T^{2}$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} - 32$$
$19$ $$T^{2} + 4T - 4$$
$23$ $$T^{2} - 4T - 28$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} - 32$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} + 2T - 71$$
$43$ $$T^{2} + 10T + 23$$
$47$ $$T^{2} + 18T + 79$$
$53$ $$T^{2} + 4T - 28$$
$59$ $$T^{2} - 128$$
$61$ $$(T - 1)^{2}$$
$67$ $$T^{2} - 10T + 23$$
$71$ $$(T - 4)^{2}$$
$73$ $$(T - 4)^{2}$$
$79$ $$T^{2} - 12T - 36$$
$83$ $$T^{2} - 4T - 124$$
$89$ $$T^{2} + 6T - 23$$
$97$ $$T^{2} + 12T - 92$$