# Properties

 Label 4680.2.a.bj Level $4680$ Weight $2$ Character orbit 4680.a Self dual yes Analytic conductor $37.370$ 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] = [4680,2,Mod(1,4680)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4680.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: $$37.3699881460$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1016.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x + 2$$ x^3 - x^2 - 6*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + q^{5} - \beta_{2} q^{7}+O(q^{10})$$ q + q^5 - b2 * q^7 $$q + q^{5} - \beta_{2} q^{7} + (\beta_1 - 2) q^{11} - q^{13} + (\beta_{2} - 2) q^{17} + ( - \beta_{2} - \beta_1 + 2) q^{19} + ( - \beta_{2} - 2 \beta_1 + 2) q^{23} + q^{25} + (\beta_{2} + \beta_1) q^{29} - 2 q^{31} - \beta_{2} q^{35} + \beta_1 q^{37} + \beta_1 q^{41} + (2 \beta_1 + 4) q^{43} + (2 \beta_1 + 4) q^{47} + ( - 2 \beta_{2} - 3 \beta_1 + 11) q^{49} + (\beta_1 + 6) q^{53} + (\beta_1 - 2) q^{55} - 4 q^{59} + (\beta_1 + 4) q^{61} - q^{65} + (2 \beta_1 + 2) q^{67} + ( - 2 \beta_{2} - \beta_1 + 2) q^{71} + (\beta_{2} - \beta_1 + 2) q^{73} + (2 \beta_{2} - 3 \beta_1 + 6) q^{77} + (2 \beta_{2} - \beta_1 - 2) q^{79} - 4 q^{83} + (\beta_{2} - 2) q^{85} + ( - 2 \beta_{2} - 3 \beta_1 + 8) q^{89} + \beta_{2} q^{91} + ( - \beta_{2} - \beta_1 + 2) q^{95} + ( - \beta_{2} + 8) q^{97}+O(q^{100})$$ q + q^5 - b2 * q^7 + (b1 - 2) * q^11 - q^13 + (b2 - 2) * q^17 + (-b2 - b1 + 2) * q^19 + (-b2 - 2*b1 + 2) * q^23 + q^25 + (b2 + b1) * q^29 - 2 * q^31 - b2 * q^35 + b1 * q^37 + b1 * q^41 + (2*b1 + 4) * q^43 + (2*b1 + 4) * q^47 + (-2*b2 - 3*b1 + 11) * q^49 + (b1 + 6) * q^53 + (b1 - 2) * q^55 - 4 * q^59 + (b1 + 4) * q^61 - q^65 + (2*b1 + 2) * q^67 + (-2*b2 - b1 + 2) * q^71 + (b2 - b1 + 2) * q^73 + (2*b2 - 3*b1 + 6) * q^77 + (2*b2 - b1 - 2) * q^79 - 4 * q^83 + (b2 - 2) * q^85 + (-2*b2 - 3*b1 + 8) * q^89 + b2 * q^91 + (-b2 - b1 + 2) * q^95 + (-b2 + 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{5} - q^{7}+O(q^{10})$$ 3 * q + 3 * q^5 - q^7 $$3 q + 3 q^{5} - q^{7} - 5 q^{11} - 3 q^{13} - 5 q^{17} + 4 q^{19} + 3 q^{23} + 3 q^{25} + 2 q^{29} - 6 q^{31} - q^{35} + q^{37} + q^{41} + 14 q^{43} + 14 q^{47} + 28 q^{49} + 19 q^{53} - 5 q^{55} - 12 q^{59} + 13 q^{61} - 3 q^{65} + 8 q^{67} + 3 q^{71} + 6 q^{73} + 17 q^{77} - 5 q^{79} - 12 q^{83} - 5 q^{85} + 19 q^{89} + q^{91} + 4 q^{95} + 23 q^{97}+O(q^{100})$$ 3 * q + 3 * q^5 - q^7 - 5 * q^11 - 3 * q^13 - 5 * q^17 + 4 * q^19 + 3 * q^23 + 3 * q^25 + 2 * q^29 - 6 * q^31 - q^35 + q^37 + q^41 + 14 * q^43 + 14 * q^47 + 28 * q^49 + 19 * q^53 - 5 * q^55 - 12 * q^59 + 13 * q^61 - 3 * q^65 + 8 * q^67 + 3 * q^71 + 6 * q^73 + 17 * q^77 - 5 * q^79 - 12 * q^83 - 5 * q^85 + 19 * q^89 + q^91 + 4 * q^95 + 23 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2\nu + 4$$ -v^2 + 2*v + 4
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 4$$ 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
 0.321637 2.85577 −2.17741
0 0 0 1.00000 0 −4.53982 0 0 0
1.2 0 0 0 1.00000 0 −1.55611 0 0 0
1.3 0 0 0 1.00000 0 5.09593 0 0 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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$13$$ $$1$$

## 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 4680.2.a.bj yes 3
3.b odd 2 1 4680.2.a.bg 3
4.b odd 2 1 9360.2.a.de 3
12.b even 2 1 9360.2.a.cz 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4680.2.a.bg 3 3.b odd 2 1
4680.2.a.bj yes 3 1.a even 1 1 trivial
9360.2.a.cz 3 12.b even 2 1
9360.2.a.de 3 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4680))$$:

 $$T_{7}^{3} + T_{7}^{2} - 24T_{7} - 36$$ T7^3 + T7^2 - 24*T7 - 36 $$T_{11}^{3} + 5T_{11}^{2} - 8T_{11} - 16$$ T11^3 + 5*T11^2 - 8*T11 - 16 $$T_{17}^{3} + 5T_{17}^{2} - 16T_{17} - 8$$ T17^3 + 5*T17^2 - 16*T17 - 8 $$T_{19}^{3} - 4T_{19}^{2} - 20T_{19} + 32$$ T19^3 - 4*T19^2 - 20*T19 + 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} + T^{2} + \cdots - 36$$
$11$ $$T^{3} + 5 T^{2} + \cdots - 16$$
$13$ $$(T + 1)^{3}$$
$17$ $$T^{3} + 5 T^{2} + \cdots - 8$$
$19$ $$T^{3} - 4 T^{2} + \cdots + 32$$
$23$ $$T^{3} - 3 T^{2} + \cdots + 232$$
$29$ $$T^{3} - 2 T^{2} + \cdots + 16$$
$31$ $$(T + 2)^{3}$$
$37$ $$T^{3} - T^{2} + \cdots + 12$$
$41$ $$T^{3} - T^{2} + \cdots + 12$$
$43$ $$T^{3} - 14T^{2} + 256$$
$47$ $$T^{3} - 14T^{2} + 256$$
$53$ $$T^{3} - 19 T^{2} + \cdots - 144$$
$59$ $$(T + 4)^{3}$$
$61$ $$T^{3} - 13 T^{2} + \cdots - 4$$
$67$ $$T^{3} - 8 T^{2} + \cdots + 208$$
$71$ $$T^{3} - 3 T^{2} + \cdots - 192$$
$73$ $$T^{3} - 6 T^{2} + \cdots - 24$$
$79$ $$T^{3} + 5 T^{2} + \cdots - 432$$
$83$ $$(T + 4)^{3}$$
$89$ $$T^{3} - 19 T^{2} + \cdots + 1284$$
$97$ $$T^{3} - 23 T^{2} + \cdots - 292$$