# Properties

 Label 4140.2.a.n Level $4140$ Weight $2$ Character orbit 4140.a Self dual yes Analytic conductor $33.058$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

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: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1380) 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + \beta q^{7}+O(q^{10})$$ q - q^5 + b * q^7 $$q - q^{5} + \beta q^{7} - 2 \beta q^{11} + 2 q^{13} + (\beta - 2) q^{17} - 2 q^{19} - q^{23} + q^{25} + (3 \beta - 2) q^{29} - 3 \beta q^{31} - \beta q^{35} + (\beta + 4) q^{37} + ( - 3 \beta - 2) q^{41} - 4 q^{47} + (\beta - 3) q^{49} + (\beta - 6) q^{53} + 2 \beta q^{55} + (\beta - 2) q^{59} + ( - 2 \beta + 6) q^{61} - 2 q^{65} + ( - 3 \beta + 4) q^{67} + ( - \beta - 2) q^{71} + (4 \beta + 2) q^{73} + ( - 2 \beta - 8) q^{77} + ( - 2 \beta - 6) q^{79} + (3 \beta - 8) q^{83} + ( - \beta + 2) q^{85} + ( - 2 \beta + 2) q^{89} + 2 \beta q^{91} + 2 q^{95} + ( - 2 \beta + 8) q^{97} +O(q^{100})$$ q - q^5 + b * q^7 - 2*b * q^11 + 2 * q^13 + (b - 2) * q^17 - 2 * q^19 - q^23 + q^25 + (3*b - 2) * q^29 - 3*b * q^31 - b * q^35 + (b + 4) * q^37 + (-3*b - 2) * q^41 - 4 * q^47 + (b - 3) * q^49 + (b - 6) * q^53 + 2*b * q^55 + (b - 2) * q^59 + (-2*b + 6) * q^61 - 2 * q^65 + (-3*b + 4) * q^67 + (-b - 2) * q^71 + (4*b + 2) * q^73 + (-2*b - 8) * q^77 + (-2*b - 6) * q^79 + (3*b - 8) * q^83 + (-b + 2) * q^85 + (-2*b + 2) * q^89 + 2*b * q^91 + 2 * q^95 + (-2*b + 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 + q^7 $$2 q - 2 q^{5} + q^{7} - 2 q^{11} + 4 q^{13} - 3 q^{17} - 4 q^{19} - 2 q^{23} + 2 q^{25} - q^{29} - 3 q^{31} - q^{35} + 9 q^{37} - 7 q^{41} - 8 q^{47} - 5 q^{49} - 11 q^{53} + 2 q^{55} - 3 q^{59} + 10 q^{61} - 4 q^{65} + 5 q^{67} - 5 q^{71} + 8 q^{73} - 18 q^{77} - 14 q^{79} - 13 q^{83} + 3 q^{85} + 2 q^{89} + 2 q^{91} + 4 q^{95} + 14 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + q^7 - 2 * q^11 + 4 * q^13 - 3 * q^17 - 4 * q^19 - 2 * q^23 + 2 * q^25 - q^29 - 3 * q^31 - q^35 + 9 * q^37 - 7 * q^41 - 8 * q^47 - 5 * q^49 - 11 * q^53 + 2 * q^55 - 3 * q^59 + 10 * q^61 - 4 * q^65 + 5 * q^67 - 5 * q^71 + 8 * q^73 - 18 * q^77 - 14 * q^79 - 13 * q^83 + 3 * q^85 + 2 * q^89 + 2 * q^91 + 4 * q^95 + 14 * 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
 −1.56155 2.56155
0 0 0 −1.00000 0 −1.56155 0 0 0
1.2 0 0 0 −1.00000 0 2.56155 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.n 2
3.b odd 2 1 1380.2.a.g 2
12.b even 2 1 5520.2.a.br 2
15.d odd 2 1 6900.2.a.u 2
15.e even 4 2 6900.2.f.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.g 2 3.b odd 2 1
4140.2.a.n 2 1.a even 1 1 trivial
5520.2.a.br 2 12.b even 2 1
6900.2.a.u 2 15.d odd 2 1
6900.2.f.p 4 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4140))$$:

 $$T_{7}^{2} - T_{7} - 4$$ T7^2 - T7 - 4 $$T_{11}^{2} + 2T_{11} - 16$$ T11^2 + 2*T11 - 16 $$T_{13} - 2$$ T13 - 2 $$T_{17}^{2} + 3T_{17} - 2$$ T17^2 + 3*T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - T - 4$$
$11$ $$T^{2} + 2T - 16$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} + 3T - 2$$
$19$ $$(T + 2)^{2}$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} + T - 38$$
$31$ $$T^{2} + 3T - 36$$
$37$ $$T^{2} - 9T + 16$$
$41$ $$T^{2} + 7T - 26$$
$43$ $$T^{2}$$
$47$ $$(T + 4)^{2}$$
$53$ $$T^{2} + 11T + 26$$
$59$ $$T^{2} + 3T - 2$$
$61$ $$T^{2} - 10T + 8$$
$67$ $$T^{2} - 5T - 32$$
$71$ $$T^{2} + 5T + 2$$
$73$ $$T^{2} - 8T - 52$$
$79$ $$T^{2} + 14T + 32$$
$83$ $$T^{2} + 13T + 4$$
$89$ $$T^{2} - 2T - 16$$
$97$ $$T^{2} - 14T + 32$$
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