# Properties

 Label 4140.2.a.l Level $4140$ Weight $2$ Character orbit 4140.a Self dual yes Analytic conductor $33.058$ 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] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, 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("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4140.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: $$33.0580664368$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + (\beta - 1) q^{7}+O(q^{10})$$ q - q^5 + (b - 1) * q^7 $$q - q^{5} + (\beta - 1) q^{7} + \beta q^{11} + ( - \beta + 2) q^{13} + ( - \beta + 3) q^{17} + ( - \beta - 4) q^{19} + q^{23} + q^{25} + ( - 2 \beta - 3) q^{29} - 7 q^{31} + ( - \beta + 1) q^{35} + ( - 3 \beta - 1) q^{37} + ( - 2 \beta + 3) q^{41} + (2 \beta - 4) q^{43} + \beta q^{47} - 2 \beta q^{49} + (3 \beta + 3) q^{53} - \beta q^{55} + (2 \beta + 3) q^{59} + (3 \beta + 2) q^{61} + (\beta - 2) q^{65} + ( - \beta - 1) q^{67} + 3 q^{71} + ( - 3 \beta + 2) q^{73} + ( - \beta + 6) q^{77} - 4 q^{79} + (3 \beta - 3) q^{83} + (\beta - 3) q^{85} + (2 \beta - 12) q^{89} + (3 \beta - 8) q^{91} + (\beta + 4) q^{95} + (2 \beta - 10) q^{97} +O(q^{100})$$ q - q^5 + (b - 1) * q^7 + b * q^11 + (-b + 2) * q^13 + (-b + 3) * q^17 + (-b - 4) * q^19 + q^23 + q^25 + (-2*b - 3) * q^29 - 7 * q^31 + (-b + 1) * q^35 + (-3*b - 1) * q^37 + (-2*b + 3) * q^41 + (2*b - 4) * q^43 + b * q^47 - 2*b * q^49 + (3*b + 3) * q^53 - b * q^55 + (2*b + 3) * q^59 + (3*b + 2) * q^61 + (b - 2) * q^65 + (-b - 1) * q^67 + 3 * q^71 + (-3*b + 2) * q^73 + (-b + 6) * q^77 - 4 * q^79 + (3*b - 3) * q^83 + (b - 3) * q^85 + (2*b - 12) * q^89 + (3*b - 8) * q^91 + (b + 4) * q^95 + (2*b - 10) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 - 2 * q^7 $$2 q - 2 q^{5} - 2 q^{7} + 4 q^{13} + 6 q^{17} - 8 q^{19} + 2 q^{23} + 2 q^{25} - 6 q^{29} - 14 q^{31} + 2 q^{35} - 2 q^{37} + 6 q^{41} - 8 q^{43} + 6 q^{53} + 6 q^{59} + 4 q^{61} - 4 q^{65} - 2 q^{67} + 6 q^{71} + 4 q^{73} + 12 q^{77} - 8 q^{79} - 6 q^{83} - 6 q^{85} - 24 q^{89} - 16 q^{91} + 8 q^{95} - 20 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 2 * q^7 + 4 * q^13 + 6 * q^17 - 8 * q^19 + 2 * q^23 + 2 * q^25 - 6 * q^29 - 14 * q^31 + 2 * q^35 - 2 * q^37 + 6 * q^41 - 8 * q^43 + 6 * q^53 + 6 * q^59 + 4 * q^61 - 4 * q^65 - 2 * q^67 + 6 * q^71 + 4 * q^73 + 12 * q^77 - 8 * q^79 - 6 * q^83 - 6 * q^85 - 24 * q^89 - 16 * q^91 + 8 * q^95 - 20 * 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
 −2.44949 2.44949
0 0 0 −1.00000 0 −3.44949 0 0 0
1.2 0 0 0 −1.00000 0 1.44949 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
$$2$$ $$-1$$
$$3$$ $$+1$$
$$5$$ $$+1$$
$$23$$ $$-1$$

## 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 4140.2.a.l 2
3.b odd 2 1 4140.2.a.q yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4140.2.a.l 2 1.a even 1 1 trivial
4140.2.a.q yes 2 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4140))$$:

 $$T_{7}^{2} + 2T_{7} - 5$$ T7^2 + 2*T7 - 5 $$T_{11}^{2} - 6$$ T11^2 - 6 $$T_{13}^{2} - 4T_{13} - 2$$ T13^2 - 4*T13 - 2 $$T_{17}^{2} - 6T_{17} + 3$$ T17^2 - 6*T17 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 2T - 5$$
$11$ $$T^{2} - 6$$
$13$ $$T^{2} - 4T - 2$$
$17$ $$T^{2} - 6T + 3$$
$19$ $$T^{2} + 8T + 10$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} + 6T - 15$$
$31$ $$(T + 7)^{2}$$
$37$ $$T^{2} + 2T - 53$$
$41$ $$T^{2} - 6T - 15$$
$43$ $$T^{2} + 8T - 8$$
$47$ $$T^{2} - 6$$
$53$ $$T^{2} - 6T - 45$$
$59$ $$T^{2} - 6T - 15$$
$61$ $$T^{2} - 4T - 50$$
$67$ $$T^{2} + 2T - 5$$
$71$ $$(T - 3)^{2}$$
$73$ $$T^{2} - 4T - 50$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 6T - 45$$
$89$ $$T^{2} + 24T + 120$$
$97$ $$T^{2} + 20T + 76$$