# Properties

 Label 4140.2.a.t Level $4140$ Weight $2$ Character orbit 4140.a Self dual yes Analytic conductor $33.058$ Analytic rank $0$ Dimension $5$ 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: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.14345904.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 13x^{3} + 34x^{2} - 11x - 12$$ x^5 - 2*x^4 - 13*x^3 + 34*x^2 - 11*x - 12 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + (\beta_{3} + 1) q^{7}+O(q^{10})$$ q - q^5 + (b3 + 1) * q^7 $$q - q^{5} + (\beta_{3} + 1) q^{7} + (\beta_{4} - 1) q^{11} + (\beta_{3} - \beta_{2} + 1) q^{13} + (\beta_{2} - \beta_1 - 1) q^{17} + (\beta_{4} + 1) q^{19} - q^{23} + q^{25} - \beta_{2} q^{29} + ( - 2 \beta_{4} + \beta_{3} - \beta_1 + 2) q^{31} + ( - \beta_{3} - 1) q^{35} + (\beta_{4} + \beta_{2} + 1) q^{37} + (\beta_{4} + \beta_{3} + 2 \beta_1) q^{41} + (\beta_{4} - \beta_{3} - \beta_{2} + \cdots + 4) q^{43}+ \cdots + (\beta_{4} - \beta_{3} + \cdots - 2 \beta_1) q^{97}+O(q^{100})$$ q - q^5 + (b3 + 1) * q^7 + (b4 - 1) * q^11 + (b3 - b2 + 1) * q^13 + (b2 - b1 - 1) * q^17 + (b4 + 1) * q^19 - q^23 + q^25 - b2 * q^29 + (-2*b4 + b3 - b1 + 2) * q^31 + (-b3 - 1) * q^35 + (b4 + b2 + 1) * q^37 + (b4 + b3 + 2*b1) * q^41 + (b4 - b3 - b2 + 2*b1 + 4) * q^43 + (-2*b4 + b3 + b2 + 1) * q^47 + (-b4 + 2*b3 + b2 + b1 + 3) * q^49 + (b4 - b3 - 2*b2 + b1 + 1) * q^53 + (-b4 + 1) * q^55 + (-2*b4 + 2*b3 - b2) * q^59 + (-b4 + 2*b2 - 2*b1 + 1) * q^61 + (-b3 + b2 - 1) * q^65 + (b4 - 2*b3 - b2 + 3) * q^67 + (-b4 + b3 - 2*b1 - 2) * q^71 + (-b3 + b2 + 3) * q^73 + (2*b4 - 3*b3 - b2 + 2*b1 + 3) * q^77 + (-2*b3 - 2*b1 + 2) * q^79 + (-4*b3 + b2 - b1 - 3) * q^83 + (-b2 + b1 + 1) * q^85 + (2*b3 + 4) * q^89 + (b4 - 2*b3 + 2*b1 + 7) * q^91 + (-b4 - 1) * q^95 + (b4 - b3 + b2 - 2*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 5 q^{5} + 4 q^{7}+O(q^{10})$$ 5 * q - 5 * q^5 + 4 * q^7 $$5 q - 5 q^{5} + 4 q^{7} - 4 q^{11} + 2 q^{13} - 2 q^{17} + 6 q^{19} - 5 q^{23} + 5 q^{25} - 2 q^{29} + 8 q^{31} - 4 q^{35} + 8 q^{37} - 2 q^{41} + 18 q^{43} + 4 q^{47} + 13 q^{49} + 2 q^{53} + 4 q^{55} - 6 q^{59} + 10 q^{61} - 2 q^{65} + 16 q^{67} - 10 q^{71} + 18 q^{73} + 16 q^{77} + 14 q^{79} - 8 q^{83} + 2 q^{85} + 18 q^{89} + 36 q^{91} - 6 q^{95} + 6 q^{97}+O(q^{100})$$ 5 * q - 5 * q^5 + 4 * q^7 - 4 * q^11 + 2 * q^13 - 2 * q^17 + 6 * q^19 - 5 * q^23 + 5 * q^25 - 2 * q^29 + 8 * q^31 - 4 * q^35 + 8 * q^37 - 2 * q^41 + 18 * q^43 + 4 * q^47 + 13 * q^49 + 2 * q^53 + 4 * q^55 - 6 * q^59 + 10 * q^61 - 2 * q^65 + 16 * q^67 - 10 * q^71 + 18 * q^73 + 16 * q^77 + 14 * q^79 - 8 * q^83 + 2 * q^85 + 18 * q^89 + 36 * q^91 - 6 * q^95 + 6 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 13x^{3} + 34x^{2} - 11x - 12$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{4} + 12\nu^{2} - 9\nu - 2 ) / 2$$ (-v^4 + 12*v^2 - 9*v - 2) / 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{4} + \nu^{3} + 13\nu^{2} - 18\nu ) / 2$$ (-v^4 + v^3 + 13*v^2 - 18*v) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - 14\nu^{2} + 7\nu + 14 ) / 2$$ (v^4 - 14*v^2 + 7*v + 14) / 2 $$\beta_{4}$$ $$=$$ $$( 2\nu^{4} - \nu^{3} - 27\nu^{2} + 29\nu + 14 ) / 2$$ (2*v^4 - v^3 - 27*v^2 + 29*v + 14) / 2
 $$\nu$$ $$=$$ $$( \beta_{4} - \beta_{3} + \beta_{2} ) / 2$$ (b4 - b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{4} - \beta_{3} - \beta_{2} - 2\beta _1 + 12 ) / 2$$ (-b4 - b3 - b2 - 2*b1 + 12) / 2 $$\nu^{3}$$ $$=$$ $$5\beta_{4} - 4\beta_{3} + 7\beta_{2} - \beta _1 - 8$$ 5*b4 - 4*b3 + 7*b2 - b1 - 8 $$\nu^{4}$$ $$=$$ $$( -21\beta_{4} - 3\beta_{3} - 21\beta_{2} - 28\beta _1 + 140 ) / 2$$ (-21*b4 - 3*b3 - 21*b2 - 28*b1 + 140) / 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
 1.84503 3.18817 −3.83694 1.23445 −0.430705
0 0 0 −1.00000 0 −3.57723 0 0 0
1.2 0 0 0 −1.00000 0 −0.334658 0 0 0
1.3 0 0 0 −1.00000 0 −0.113901 0 0 0
1.4 0 0 0 −1.00000 0 2.81459 0 0 0
1.5 0 0 0 −1.00000 0 5.21120 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.t 5
3.b odd 2 1 4140.2.a.u yes 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4140.2.a.t 5 1.a even 1 1 trivial
4140.2.a.u yes 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4140))$$:

 $$T_{7}^{5} - 4T_{7}^{4} - 16T_{7}^{3} + 46T_{7}^{2} + 23T_{7} + 2$$ T7^5 - 4*T7^4 - 16*T7^3 + 46*T7^2 + 23*T7 + 2 $$T_{11}^{5} + 4T_{11}^{4} - 26T_{11}^{3} - 108T_{11}^{2} + 84T_{11} + 432$$ T11^5 + 4*T11^4 - 26*T11^3 - 108*T11^2 + 84*T11 + 432 $$T_{13}^{5} - 2T_{13}^{4} - 50T_{13}^{3} + 88T_{13}^{2} + 372T_{13} - 72$$ T13^5 - 2*T13^4 - 50*T13^3 + 88*T13^2 + 372*T13 - 72 $$T_{17}^{5} + 2T_{17}^{4} - 44T_{17}^{3} - 156T_{17}^{2} + 119T_{17} + 666$$ T17^5 + 2*T17^4 - 44*T17^3 - 156*T17^2 + 119*T17 + 666

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5}$$
$5$ $$(T + 1)^{5}$$
$7$ $$T^{5} - 4 T^{4} + \cdots + 2$$
$11$ $$T^{5} + 4 T^{4} + \cdots + 432$$
$13$ $$T^{5} - 2 T^{4} + \cdots - 72$$
$17$ $$T^{5} + 2 T^{4} + \cdots + 666$$
$19$ $$T^{5} - 6 T^{4} + \cdots + 72$$
$23$ $$(T + 1)^{5}$$
$29$ $$T^{5} + 2 T^{4} + \cdots + 252$$
$31$ $$T^{5} - 8 T^{4} + \cdots - 112$$
$37$ $$T^{5} - 8 T^{4} + \cdots + 122$$
$41$ $$T^{5} + 2 T^{4} + \cdots - 18048$$
$43$ $$T^{5} - 18 T^{4} + \cdots - 3744$$
$47$ $$T^{5} - 4 T^{4} + \cdots + 7344$$
$53$ $$T^{5} - 2 T^{4} + \cdots - 6426$$
$59$ $$T^{5} + 6 T^{4} + \cdots + 18804$$
$61$ $$T^{5} - 10 T^{4} + \cdots + 376$$
$67$ $$T^{5} - 16 T^{4} + \cdots + 27618$$
$71$ $$T^{5} + 10 T^{4} + \cdots + 816$$
$73$ $$T^{5} - 18 T^{4} + \cdots - 136$$
$79$ $$T^{5} - 14 T^{4} + \cdots - 4608$$
$83$ $$T^{5} + 8 T^{4} + \cdots + 33876$$
$89$ $$T^{5} - 18 T^{4} + \cdots + 1152$$
$97$ $$T^{5} - 6 T^{4} + \cdots - 32$$