# Properties

 Label 9680.2.a.cq Level $9680$ Weight $2$ Character orbit 9680.a Self dual yes Analytic conductor $77.295$ Analytic rank $1$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.4752.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 3x^{2} + 4x + 1$$ x^4 - 2*x^3 - 3*x^2 + 4*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{3} - q^{5} + (\beta_{3} - \beta_{2} - 2) q^{7} + (\beta_1 + 1) q^{9}+O(q^{10})$$ q + (b2 + 1) * q^3 - q^5 + (b3 - b2 - 2) * q^7 + (b1 + 1) * q^9 $$q + (\beta_{2} + 1) q^{3} - q^{5} + (\beta_{3} - \beta_{2} - 2) q^{7} + (\beta_1 + 1) q^{9} + (\beta_1 + 1) q^{13} + ( - \beta_{2} - 1) q^{15} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{19}+ \cdots + (4 \beta_{3} - 6 \beta_{2} + \beta_1 - 3) q^{97}+O(q^{100})$$ q + (b2 + 1) * q^3 - q^5 + (b3 - b2 - 2) * q^7 + (b1 + 1) * q^9 + (b1 + 1) * q^13 + (-b2 - 1) * q^15 + (-2*b3 + 2*b2 - b1 + 1) * q^19 + (b3 - 2*b2 - 4) * q^21 + (-2*b3 - b1 - 1) * q^23 + q^25 + (2*b3 - b2 + b1) * q^27 + 4 * q^29 + (2*b3 - 2*b2 - b1 - 5) * q^31 + (-b3 + b2 + 2) * q^35 + (-2*b1 - 2) * q^37 + (2*b3 + 2*b2 + b1 + 3) * q^39 + (-3*b3 + 2*b2 - b1 + 3) * q^41 + (-b3 + 3*b2 - 2*b1 + 4) * q^43 + (-b1 - 1) * q^45 + (b2 - 2*b1 - 3) * q^47 + (-4*b3 + 4*b2 - b1 + 1) * q^49 + (-2*b2 + b1 + 1) * q^53 + (-4*b3 - b1 + 3) * q^57 + (-2*b3 - 2*b2 - b1 - 1) * q^59 + (3*b3 - 4*b2 + b1 - 3) * q^61 + (-2*b3 - b1 - 3) * q^63 + (-b1 - 1) * q^65 + (-3*b2 + 2*b1 - 1) * q^67 + (-4*b3 - 3*b1 - 5) * q^69 + (6*b3 - 2*b2 + b1 - 3) * q^71 + 8 * q^73 + (b2 + 1) * q^75 + (-2*b3 + 2*b2 + 2*b1 + 2) * q^79 + (4*b3 - b1 - 2) * q^81 + (2*b3 + 4*b2 + b1 - 1) * q^83 + (4*b2 + 4) * q^87 + (4*b3 - 4*b2 + 2*b1 + 1) * q^89 + (-2*b3 - b1 - 3) * q^91 + (-6*b2 - b1 - 11) * q^93 + (2*b3 - 2*b2 + b1 - 1) * q^95 + (4*b3 - 6*b2 + b1 - 3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 - 4 * q^5 - 6 * q^7 + 4 * q^9 $$4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9} + 4 q^{13} - 2 q^{15} - 12 q^{21} - 4 q^{23} + 4 q^{25} + 2 q^{27} + 16 q^{29} - 16 q^{31} + 6 q^{35} - 8 q^{37} + 8 q^{39} + 8 q^{41} + 10 q^{43} - 4 q^{45} - 14 q^{47} - 4 q^{49} + 8 q^{53} + 12 q^{57} - 4 q^{61} - 12 q^{63} - 4 q^{65} + 2 q^{67} - 20 q^{69} - 8 q^{71} + 32 q^{73} + 2 q^{75} + 4 q^{79} - 8 q^{81} - 12 q^{83} + 8 q^{87} + 12 q^{89} - 12 q^{91} - 32 q^{93}+O(q^{100})$$ 4 * q + 2 * q^3 - 4 * q^5 - 6 * q^7 + 4 * q^9 + 4 * q^13 - 2 * q^15 - 12 * q^21 - 4 * q^23 + 4 * q^25 + 2 * q^27 + 16 * q^29 - 16 * q^31 + 6 * q^35 - 8 * q^37 + 8 * q^39 + 8 * q^41 + 10 * q^43 - 4 * q^45 - 14 * q^47 - 4 * q^49 + 8 * q^53 + 12 * q^57 - 4 * q^61 - 12 * q^63 - 4 * q^65 + 2 * q^67 - 20 * q^69 - 8 * q^71 + 32 * q^73 + 2 * q^75 + 4 * q^79 - 8 * q^81 - 12 * q^83 + 8 * q^87 + 12 * q^89 - 12 * q^91 - 32 * q^93

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

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu$$ v^3 - v^2 - 3*v $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} + \beta _1 + 5 ) / 2$$ (2*b3 + b1 + 5) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 2\beta _1 + 4$$ b3 + 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
 1.21969 −1.49551 −0.219687 2.49551
0 −2.33225 0 −1.00000 0 −0.399804 0 2.43937 0
1.2 0 −0.0947876 0 −1.00000 0 0.826838 0 −2.99102 0
1.3 0 1.60020 0 −1.00000 0 −4.33225 0 −0.439374 0
1.4 0 2.82684 0 −1.00000 0 −2.09479 0 4.99102 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.cq 4
4.b odd 2 1 4840.2.a.x yes 4
11.b odd 2 1 9680.2.a.cr 4
44.c even 2 1 4840.2.a.w 4

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

 $$T_{3}^{4} - 2T_{3}^{3} - 6T_{3}^{2} + 10T_{3} + 1$$ T3^4 - 2*T3^3 - 6*T3^2 + 10*T3 + 1 $$T_{7}^{4} + 6T_{7}^{3} + 6T_{7}^{2} - 6T_{7} - 3$$ T7^4 + 6*T7^3 + 6*T7^2 - 6*T7 - 3 $$T_{13}^{4} - 4T_{13}^{3} - 12T_{13}^{2} + 32T_{13} + 16$$ T13^4 - 4*T13^3 - 12*T13^2 + 32*T13 + 16 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 2 T^{3} + \cdots + 1$$
$5$ $$(T + 1)^{4}$$
$7$ $$T^{4} + 6 T^{3} + \cdots - 3$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 4 T^{3} + \cdots + 16$$
$17$ $$T^{4}$$
$19$ $$T^{4} - 36 T^{2} + \cdots - 48$$
$23$ $$T^{4} + 4 T^{3} + \cdots + 16$$
$29$ $$(T - 4)^{4}$$
$31$ $$T^{4} + 16 T^{3} + \cdots - 1136$$
$37$ $$T^{4} + 8 T^{3} + \cdots + 256$$
$41$ $$T^{4} - 8 T^{3} + \cdots + 1$$
$43$ $$T^{4} - 10 T^{3} + \cdots + 121$$
$47$ $$T^{4} + 14 T^{3} + \cdots + 157$$
$53$ $$T^{4} - 8 T^{3} + \cdots + 16$$
$59$ $$T^{4} - 108 T^{2} + \cdots - 432$$
$61$ $$T^{4} + 4 T^{3} + \cdots + 229$$
$67$ $$T^{4} - 2 T^{3} + \cdots + 577$$
$71$ $$T^{4} + 8 T^{3} + \cdots + 3664$$
$73$ $$(T - 8)^{4}$$
$79$ $$T^{4} - 4 T^{3} + \cdots - 752$$
$83$ $$T^{4} + 12 T^{3} + \cdots - 4656$$
$89$ $$T^{4} - 12 T^{3} + \cdots - 4287$$
$97$ $$T^{4} - 204 T^{2} + \cdots + 3408$$