# Properties

 Label 140.2.a.b Level $140$ Weight $2$ Character orbit 140.a Self dual yes Analytic conductor $1.118$ 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] = [140,2,Mod(1,140)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$140 = 2^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 140.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: $$1.11790562830$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - q^{5} - q^{7} + 6 q^{9}+O(q^{10})$$ q + 3 * q^3 - q^5 - q^7 + 6 * q^9 $$q + 3 q^{3} - q^{5} - q^{7} + 6 q^{9} - 5 q^{11} - 3 q^{13} - 3 q^{15} - q^{17} + 6 q^{19} - 3 q^{21} + 6 q^{23} + q^{25} + 9 q^{27} - 9 q^{29} - 4 q^{31} - 15 q^{33} + q^{35} + 2 q^{37} - 9 q^{39} - 4 q^{41} + 10 q^{43} - 6 q^{45} - q^{47} + q^{49} - 3 q^{51} + 4 q^{53} + 5 q^{55} + 18 q^{57} - 8 q^{59} - 8 q^{61} - 6 q^{63} + 3 q^{65} + 12 q^{67} + 18 q^{69} + 8 q^{71} + 2 q^{73} + 3 q^{75} + 5 q^{77} + 13 q^{79} + 9 q^{81} - 4 q^{83} + q^{85} - 27 q^{87} + 4 q^{89} + 3 q^{91} - 12 q^{93} - 6 q^{95} - 13 q^{97} - 30 q^{99}+O(q^{100})$$ q + 3 * q^3 - q^5 - q^7 + 6 * q^9 - 5 * q^11 - 3 * q^13 - 3 * q^15 - q^17 + 6 * q^19 - 3 * q^21 + 6 * q^23 + q^25 + 9 * q^27 - 9 * q^29 - 4 * q^31 - 15 * q^33 + q^35 + 2 * q^37 - 9 * q^39 - 4 * q^41 + 10 * q^43 - 6 * q^45 - q^47 + q^49 - 3 * q^51 + 4 * q^53 + 5 * q^55 + 18 * q^57 - 8 * q^59 - 8 * q^61 - 6 * q^63 + 3 * q^65 + 12 * q^67 + 18 * q^69 + 8 * q^71 + 2 * q^73 + 3 * q^75 + 5 * q^77 + 13 * q^79 + 9 * q^81 - 4 * q^83 + q^85 - 27 * q^87 + 4 * q^89 + 3 * q^91 - 12 * q^93 - 6 * q^95 - 13 * q^97 - 30 * q^99

## 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 3.00000 0 −1.00000 0 −1.00000 0 6.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$

## 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 140.2.a.b 1
3.b odd 2 1 1260.2.a.h 1
4.b odd 2 1 560.2.a.a 1
5.b even 2 1 700.2.a.b 1
5.c odd 4 2 700.2.e.a 2
7.b odd 2 1 980.2.a.b 1
7.c even 3 2 980.2.i.b 2
7.d odd 6 2 980.2.i.j 2
8.b even 2 1 2240.2.a.c 1
8.d odd 2 1 2240.2.a.bb 1
12.b even 2 1 5040.2.a.bd 1
15.d odd 2 1 6300.2.a.bf 1
15.e even 4 2 6300.2.k.p 2
20.d odd 2 1 2800.2.a.be 1
20.e even 4 2 2800.2.g.c 2
21.c even 2 1 8820.2.a.n 1
28.d even 2 1 3920.2.a.bl 1
35.c odd 2 1 4900.2.a.u 1
35.f even 4 2 4900.2.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.b 1 1.a even 1 1 trivial
560.2.a.a 1 4.b odd 2 1
700.2.a.b 1 5.b even 2 1
700.2.e.a 2 5.c odd 4 2
980.2.a.b 1 7.b odd 2 1
980.2.i.b 2 7.c even 3 2
980.2.i.j 2 7.d odd 6 2
1260.2.a.h 1 3.b odd 2 1
2240.2.a.c 1 8.b even 2 1
2240.2.a.bb 1 8.d odd 2 1
2800.2.a.be 1 20.d odd 2 1
2800.2.g.c 2 20.e even 4 2
3920.2.a.bl 1 28.d even 2 1
4900.2.a.u 1 35.c odd 2 1
4900.2.e.a 2 35.f even 4 2
5040.2.a.bd 1 12.b even 2 1
6300.2.a.bf 1 15.d odd 2 1
6300.2.k.p 2 15.e even 4 2
8820.2.a.n 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(140))$$.

## Hecke characteristic polynomials

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