Properties

 Label 4140.2.a.i 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 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)

 $$f(q)$$ $$=$$ $$q + q^{5} - q^{7}+O(q^{10})$$ q + q^5 - q^7 $$q + q^{5} - q^{7} - 4 q^{13} + 3 q^{17} - 4 q^{19} + q^{23} + q^{25} + 3 q^{29} - 7 q^{31} - q^{35} + 11 q^{37} + 9 q^{41} - 4 q^{43} - 6 q^{47} - 6 q^{49} - 9 q^{53} - 3 q^{59} - 10 q^{61} - 4 q^{65} - 13 q^{67} - 9 q^{71} - 16 q^{73} + 8 q^{79} + 15 q^{83} + 3 q^{85} + 4 q^{91} - 4 q^{95} + 2 q^{97}+O(q^{100})$$ q + q^5 - q^7 - 4 * q^13 + 3 * q^17 - 4 * q^19 + q^23 + q^25 + 3 * q^29 - 7 * q^31 - q^35 + 11 * q^37 + 9 * q^41 - 4 * q^43 - 6 * q^47 - 6 * q^49 - 9 * q^53 - 3 * q^59 - 10 * q^61 - 4 * q^65 - 13 * q^67 - 9 * q^71 - 16 * q^73 + 8 * q^79 + 15 * q^83 + 3 * q^85 + 4 * 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 −1.00000 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.i 1
3.b odd 2 1 1380.2.a.c 1
12.b even 2 1 5520.2.a.g 1
15.d odd 2 1 6900.2.a.b 1
15.e even 4 2 6900.2.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.c 1 3.b odd 2 1
4140.2.a.i 1 1.a even 1 1 trivial
5520.2.a.g 1 12.b even 2 1
6900.2.a.b 1 15.d odd 2 1
6900.2.f.e 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4140))$$:

 $$T_{7} + 1$$ T7 + 1 $$T_{11}$$ T11 $$T_{13} + 4$$ T13 + 4 $$T_{17} - 3$$ T17 - 3

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 1$$
$11$ $$T$$
$13$ $$T + 4$$
$17$ $$T - 3$$
$19$ $$T + 4$$
$23$ $$T - 1$$
$29$ $$T - 3$$
$31$ $$T + 7$$
$37$ $$T - 11$$
$41$ $$T - 9$$
$43$ $$T + 4$$
$47$ $$T + 6$$
$53$ $$T + 9$$
$59$ $$T + 3$$
$61$ $$T + 10$$
$67$ $$T + 13$$
$71$ $$T + 9$$
$73$ $$T + 16$$
$79$ $$T - 8$$
$83$ $$T - 15$$
$89$ $$T$$
$97$ $$T - 2$$