# Properties

 Label 4140.2.a.f Level $4140$ Weight $2$ Character orbit 4140.a Self dual yes Analytic conductor $33.058$ Analytic rank $0$ 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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{5} - 4 q^{7}+O(q^{10})$$ q + q^5 - 4 * q^7 $$q + q^{5} - 4 q^{7} + 6 q^{11} - q^{13} + 2 q^{19} - q^{23} + q^{25} - 9 q^{29} + 5 q^{31} - 4 q^{35} + 2 q^{37} + 9 q^{41} - 4 q^{43} + 3 q^{47} + 9 q^{49} + 6 q^{53} + 6 q^{55} + 2 q^{61} - q^{65} - 10 q^{67} + 3 q^{71} - 7 q^{73} - 24 q^{77} - 10 q^{79} + 12 q^{83} + 4 q^{91} + 2 q^{95} + 8 q^{97}+O(q^{100})$$ q + q^5 - 4 * q^7 + 6 * q^11 - q^13 + 2 * q^19 - q^23 + q^25 - 9 * q^29 + 5 * q^31 - 4 * q^35 + 2 * q^37 + 9 * q^41 - 4 * q^43 + 3 * q^47 + 9 * q^49 + 6 * q^53 + 6 * q^55 + 2 * q^61 - q^65 - 10 * q^67 + 3 * q^71 - 7 * q^73 - 24 * q^77 - 10 * q^79 + 12 * q^83 + 4 * q^91 + 2 * q^95 + 8 * 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 −4.00000 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.f 1
3.b odd 2 1 460.2.a.c 1
12.b even 2 1 1840.2.a.c 1
15.d odd 2 1 2300.2.a.d 1
15.e even 4 2 2300.2.c.d 2
24.f even 2 1 7360.2.a.v 1
24.h odd 2 1 7360.2.a.i 1
60.h even 2 1 9200.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.a.c 1 3.b odd 2 1
1840.2.a.c 1 12.b even 2 1
2300.2.a.d 1 15.d odd 2 1
2300.2.c.d 2 15.e even 4 2
4140.2.a.f 1 1.a even 1 1 trivial
7360.2.a.i 1 24.h odd 2 1
7360.2.a.v 1 24.f even 2 1
9200.2.a.y 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4140))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11} - 6$$ T11 - 6 $$T_{13} + 1$$ T13 + 1 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 4$$
$11$ $$T - 6$$
$13$ $$T + 1$$
$17$ $$T$$
$19$ $$T - 2$$
$23$ $$T + 1$$
$29$ $$T + 9$$
$31$ $$T - 5$$
$37$ $$T - 2$$
$41$ $$T - 9$$
$43$ $$T + 4$$
$47$ $$T - 3$$
$53$ $$T - 6$$
$59$ $$T$$
$61$ $$T - 2$$
$67$ $$T + 10$$
$71$ $$T - 3$$
$73$ $$T + 7$$
$79$ $$T + 10$$
$83$ $$T - 12$$
$89$ $$T$$
$97$ $$T - 8$$