# Properties

 Label 140.2.a.a 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 + q^5 + q^7 - 2 * q^9 $$q + q^{3} + q^{5} + q^{7} - 2 q^{9} + 3 q^{11} - q^{13} + q^{15} - 3 q^{17} + 2 q^{19} + q^{21} - 6 q^{23} + q^{25} - 5 q^{27} - 9 q^{29} + 8 q^{31} + 3 q^{33} + q^{35} - 10 q^{37} - q^{39} + 2 q^{43} - 2 q^{45} - 3 q^{47} + q^{49} - 3 q^{51} + 3 q^{55} + 2 q^{57} + 12 q^{59} + 8 q^{61} - 2 q^{63} - q^{65} + 8 q^{67} - 6 q^{69} + 14 q^{73} + q^{75} + 3 q^{77} + 5 q^{79} + q^{81} - 12 q^{83} - 3 q^{85} - 9 q^{87} + 12 q^{89} - q^{91} + 8 q^{93} + 2 q^{95} + 17 q^{97} - 6 q^{99}+O(q^{100})$$ q + q^3 + q^5 + q^7 - 2 * q^9 + 3 * q^11 - q^13 + q^15 - 3 * q^17 + 2 * q^19 + q^21 - 6 * q^23 + q^25 - 5 * q^27 - 9 * q^29 + 8 * q^31 + 3 * q^33 + q^35 - 10 * q^37 - q^39 + 2 * q^43 - 2 * q^45 - 3 * q^47 + q^49 - 3 * q^51 + 3 * q^55 + 2 * q^57 + 12 * q^59 + 8 * q^61 - 2 * q^63 - q^65 + 8 * q^67 - 6 * q^69 + 14 * q^73 + q^75 + 3 * q^77 + 5 * q^79 + q^81 - 12 * q^83 - 3 * q^85 - 9 * q^87 + 12 * q^89 - q^91 + 8 * q^93 + 2 * q^95 + 17 * q^97 - 6 * 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 1.00000 0 1.00000 0 1.00000 0 −2.00000 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.a 1
3.b odd 2 1 1260.2.a.c 1
4.b odd 2 1 560.2.a.c 1
5.b even 2 1 700.2.a.d 1
5.c odd 4 2 700.2.e.c 2
7.b odd 2 1 980.2.a.c 1
7.c even 3 2 980.2.i.d 2
7.d odd 6 2 980.2.i.h 2
8.b even 2 1 2240.2.a.g 1
8.d odd 2 1 2240.2.a.r 1
12.b even 2 1 5040.2.a.h 1
15.d odd 2 1 6300.2.a.d 1
15.e even 4 2 6300.2.k.c 2
20.d odd 2 1 2800.2.a.y 1
20.e even 4 2 2800.2.g.j 2
21.c even 2 1 8820.2.a.r 1
28.d even 2 1 3920.2.a.u 1
35.c odd 2 1 4900.2.a.p 1
35.f even 4 2 4900.2.e.l 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(140))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T - 1$$
$11$ $$T - 3$$
$13$ $$T + 1$$
$17$ $$T + 3$$
$19$ $$T - 2$$
$23$ $$T + 6$$
$29$ $$T + 9$$
$31$ $$T - 8$$
$37$ $$T + 10$$
$41$ $$T$$
$43$ $$T - 2$$
$47$ $$T + 3$$
$53$ $$T$$
$59$ $$T - 12$$
$61$ $$T - 8$$
$67$ $$T - 8$$
$71$ $$T$$
$73$ $$T - 14$$
$79$ $$T - 5$$
$83$ $$T + 12$$
$89$ $$T - 12$$
$97$ $$T - 17$$