# Properties

 Label 4140.2.a.k Level $4140$ Weight $2$ Character orbit 4140.a Self dual yes Analytic conductor $33.058$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 460) 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)

 $$f(q)$$ $$=$$ $$q + q^{5} + 2 q^{7}+O(q^{10})$$ q + q^5 + 2 * q^7 $$q + q^{5} + 2 q^{7} - 3 q^{13} - 4 q^{17} - 4 q^{19} + q^{23} + q^{25} - q^{29} + q^{31} + 2 q^{35} - 8 q^{37} - 11 q^{41} - 10 q^{43} + q^{47} - 3 q^{49} + 6 q^{53} + 8 q^{59} - 8 q^{61} - 3 q^{65} + 12 q^{67} - 13 q^{71} + 7 q^{73} - 12 q^{79} - 16 q^{83} - 4 q^{85} + 6 q^{89} - 6 q^{91} - 4 q^{95} + 2 q^{97}+O(q^{100})$$ q + q^5 + 2 * q^7 - 3 * q^13 - 4 * q^17 - 4 * q^19 + q^23 + q^25 - q^29 + q^31 + 2 * q^35 - 8 * q^37 - 11 * q^41 - 10 * q^43 + q^47 - 3 * q^49 + 6 * q^53 + 8 * q^59 - 8 * q^61 - 3 * q^65 + 12 * q^67 - 13 * q^71 + 7 * q^73 - 12 * q^79 - 16 * q^83 - 4 * q^85 + 6 * q^89 - 6 * q^91 - 4 * q^95 + 2 * 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
 0
0 0 0 1.00000 0 2.00000 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.k 1
3.b odd 2 1 460.2.a.d 1
12.b even 2 1 1840.2.a.a 1
15.d odd 2 1 2300.2.a.a 1
15.e even 4 2 2300.2.c.a 2
24.f even 2 1 7360.2.a.bb 1
24.h odd 2 1 7360.2.a.b 1
60.h even 2 1 9200.2.a.bk 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.a.d 1 3.b odd 2 1
1840.2.a.a 1 12.b even 2 1
2300.2.a.a 1 15.d odd 2 1
2300.2.c.a 2 15.e even 4 2
4140.2.a.k 1 1.a even 1 1 trivial
7360.2.a.b 1 24.h odd 2 1
7360.2.a.bb 1 24.f even 2 1
9200.2.a.bk 1 60.h even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T - 2$$
$11$ $$T$$
$13$ $$T + 3$$
$17$ $$T + 4$$
$19$ $$T + 4$$
$23$ $$T - 1$$
$29$ $$T + 1$$
$31$ $$T - 1$$
$37$ $$T + 8$$
$41$ $$T + 11$$
$43$ $$T + 10$$
$47$ $$T - 1$$
$53$ $$T - 6$$
$59$ $$T - 8$$
$61$ $$T + 8$$
$67$ $$T - 12$$
$71$ $$T + 13$$
$73$ $$T - 7$$
$79$ $$T + 12$$
$83$ $$T + 16$$
$89$ $$T - 6$$
$97$ $$T - 2$$