Properties

 Label 4680.2.a.bk 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: 3.3.621.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 3$$ x^3 - 6*x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{5} + (\beta_{2} + 1) q^{7}+O(q^{10})$$ q + q^5 + (b2 + 1) * q^7 $$q + q^{5} + (\beta_{2} + 1) q^{7} + ( - \beta_{2} - \beta_1 + 1) q^{11} + q^{13} + ( - \beta_{2} + 1) q^{17} + (\beta_1 + 2) q^{19} + ( - \beta_{2} + 1) q^{23} + q^{25} - \beta_1 q^{29} + 2 q^{31} + (\beta_{2} + 1) q^{35} + (\beta_{2} - \beta_1 + 1) q^{37} + (3 \beta_{2} + \beta_1 - 1) q^{41} + (2 \beta_{2} + 2) q^{43} + ( - 2 \beta_{2} + 2) q^{47} + (\beta_{2} - \beta_1) q^{49} + (\beta_{2} + \beta_1 - 1) q^{53} + ( - \beta_{2} - \beta_1 + 1) q^{55} + 2 \beta_1 q^{59} + ( - \beta_{2} + \beta_1 + 3) q^{61} + q^{65} + ( - 2 \beta_{2} + 4) q^{67} + (3 \beta_{2} - \beta_1 + 1) q^{71} + (2 \beta_{2} - \beta_1) q^{73} + (3 \beta_{2} - \beta_1 - 3) q^{77} + (3 \beta_{2} + \beta_1 + 5) q^{79} + ( - \beta_{2} + 1) q^{85} + ( - 3 \beta_{2} - \beta_1 - 3) q^{89} + (\beta_{2} + 1) q^{91} + (\beta_1 + 2) q^{95} + ( - 3 \beta_{2} - 2 \beta_1 + 5) q^{97}+O(q^{100})$$ q + q^5 + (b2 + 1) * q^7 + (-b2 - b1 + 1) * q^11 + q^13 + (-b2 + 1) * q^17 + (b1 + 2) * q^19 + (-b2 + 1) * q^23 + q^25 - b1 * q^29 + 2 * q^31 + (b2 + 1) * q^35 + (b2 - b1 + 1) * q^37 + (3*b2 + b1 - 1) * q^41 + (2*b2 + 2) * q^43 + (-2*b2 + 2) * q^47 + (b2 - b1) * q^49 + (b2 + b1 - 1) * q^53 + (-b2 - b1 + 1) * q^55 + 2*b1 * q^59 + (-b2 + b1 + 3) * q^61 + q^65 + (-2*b2 + 4) * q^67 + (3*b2 - b1 + 1) * q^71 + (2*b2 - b1) * q^73 + (3*b2 - b1 - 3) * q^77 + (3*b2 + b1 + 5) * q^79 + (-b2 + 1) * q^85 + (-3*b2 - b1 - 3) * q^89 + (b2 + 1) * q^91 + (b1 + 2) * q^95 + (-3*b2 - 2*b1 + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{5} + 3 q^{7}+O(q^{10})$$ 3 * q + 3 * q^5 + 3 * q^7 $$3 q + 3 q^{5} + 3 q^{7} + 3 q^{11} + 3 q^{13} + 3 q^{17} + 6 q^{19} + 3 q^{23} + 3 q^{25} + 6 q^{31} + 3 q^{35} + 3 q^{37} - 3 q^{41} + 6 q^{43} + 6 q^{47} - 3 q^{53} + 3 q^{55} + 9 q^{61} + 3 q^{65} + 12 q^{67} + 3 q^{71} - 9 q^{77} + 15 q^{79} + 3 q^{85} - 9 q^{89} + 3 q^{91} + 6 q^{95} + 15 q^{97}+O(q^{100})$$ 3 * q + 3 * q^5 + 3 * q^7 + 3 * q^11 + 3 * q^13 + 3 * q^17 + 6 * q^19 + 3 * q^23 + 3 * q^25 + 6 * q^31 + 3 * q^35 + 3 * q^37 - 3 * q^41 + 6 * q^43 + 6 * q^47 - 3 * q^53 + 3 * q^55 + 9 * q^61 + 3 * q^65 + 12 * q^67 + 3 * q^71 - 9 * q^77 + 15 * q^79 + 3 * q^85 - 9 * q^89 + 3 * q^91 + 6 * q^95 + 15 * q^97

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

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} + \beta _1 + 8 ) / 2$$ (2*b2 + b1 + 8) / 2

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
 −0.523976 2.66908 −2.14510
0 0 0 1.00000 0 −2.20147 0 0 0
1.2 0 0 0 1.00000 0 1.45490 0 0 0
1.3 0 0 0 1.00000 0 3.74657 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.bk yes 3
3.b odd 2 1 4680.2.a.bi 3
4.b odd 2 1 9360.2.a.dc 3
12.b even 2 1 9360.2.a.cx 3

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

 $$T_{7}^{3} - 3T_{7}^{2} - 6T_{7} + 12$$ T7^3 - 3*T7^2 - 6*T7 + 12 $$T_{11}^{3} - 3T_{11}^{2} - 24T_{11} + 64$$ T11^3 - 3*T11^2 - 24*T11 + 64 $$T_{17}^{3} - 3T_{17}^{2} - 6T_{17} + 4$$ T17^3 - 3*T17^2 - 6*T17 + 4 $$T_{19}^{3} - 6T_{19}^{2} - 12T_{19} + 16$$ T19^3 - 6*T19^2 - 12*T19 + 16

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - 3 T^{2} + \cdots + 12$$
$11$ $$T^{3} - 3 T^{2} + \cdots + 64$$
$13$ $$(T - 1)^{3}$$
$17$ $$T^{3} - 3 T^{2} + \cdots + 4$$
$19$ $$T^{3} - 6 T^{2} + \cdots + 16$$
$23$ $$T^{3} - 3 T^{2} + \cdots + 4$$
$29$ $$T^{3} - 24T + 24$$
$31$ $$(T - 2)^{3}$$
$37$ $$T^{3} - 3 T^{2} + \cdots - 36$$
$41$ $$T^{3} + 3 T^{2} + \cdots + 196$$
$43$ $$T^{3} - 6 T^{2} + \cdots + 96$$
$47$ $$T^{3} - 6 T^{2} + \cdots + 32$$
$53$ $$T^{3} + 3 T^{2} + \cdots - 64$$
$59$ $$T^{3} - 96T - 192$$
$61$ $$T^{3} - 9 T^{2} + \cdots + 164$$
$67$ $$T^{3} - 12 T^{2} + \cdots + 48$$
$71$ $$T^{3} - 3 T^{2} + \cdots - 304$$
$73$ $$T^{3} - 72T - 232$$
$79$ $$T^{3} - 15 T^{2} + \cdots + 592$$
$83$ $$T^{3}$$
$89$ $$T^{3} + 9 T^{2} + \cdots - 516$$
$97$ $$T^{3} - 15 T^{2} + \cdots + 628$$