# Properties

 Label 140.2.s.b Level $140$ Weight $2$ Character orbit 140.s Analytic conductor $1.118$ Analytic rank $0$ Dimension $32$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,2,Mod(19,140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 5]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("140.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$140 = 2^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 140.s (of order $$6$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$1.11790562830$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 6 q^{4} - 6 q^{5} + 4 q^{9}+O(q^{10})$$ 32 * q + 6 * q^4 - 6 * q^5 + 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 6 q^{4} - 6 q^{5} + 4 q^{9} - 12 q^{10} + 22 q^{14} + 18 q^{16} - 52 q^{21} - 48 q^{24} - 26 q^{25} - 18 q^{26} - 26 q^{30} - 28 q^{36} + 42 q^{40} - 26 q^{44} + 36 q^{45} - 22 q^{46} + 36 q^{50} + 48 q^{54} - 16 q^{56} + 4 q^{60} + 36 q^{61} + 36 q^{64} - 4 q^{65} - 24 q^{66} + 26 q^{70} + 14 q^{74} + 72 q^{80} + 72 q^{81} + 56 q^{84} + 20 q^{85} + 8 q^{86} - 108 q^{89} + 30 q^{94} + 60 q^{96}+O(q^{100})$$ 32 * q + 6 * q^4 - 6 * q^5 + 4 * q^9 - 12 * q^10 + 22 * q^14 + 18 * q^16 - 52 * q^21 - 48 * q^24 - 26 * q^25 - 18 * q^26 - 26 * q^30 - 28 * q^36 + 42 * q^40 - 26 * q^44 + 36 * q^45 - 22 * q^46 + 36 * q^50 + 48 * q^54 - 16 * q^56 + 4 * q^60 + 36 * q^61 + 36 * q^64 - 4 * q^65 - 24 * q^66 + 26 * q^70 + 14 * q^74 + 72 * q^80 + 72 * q^81 + 56 * q^84 + 20 * q^85 + 8 * q^86 - 108 * q^89 + 30 * q^94 + 60 * q^96

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.41371 + 0.0377920i 0.634715 0.366453i 1.99714 0.106854i −0.661137 + 2.13609i −0.883452 + 0.542044i 2.56107 0.664037i −2.81934 + 0.226536i −1.23142 + 2.13289i 0.853928 3.04480i
19.2 −1.38503 + 0.285823i 2.24836 1.29809i 1.83661 0.791746i 2.07525 0.832668i −2.74302 + 2.44053i −2.57589 0.603960i −2.31746 + 1.62154i 1.87009 3.23909i −2.63629 + 1.74642i
19.3 −1.36614 0.365598i −1.28100 + 0.739583i 1.73268 + 0.998916i −1.26649 1.84283i 2.02041 0.542044i 0.664037 + 2.56107i −2.00188 1.99812i −0.406034 + 0.703271i 1.05646 + 2.98058i
19.4 −1.13507 + 0.843578i −1.62820 + 0.940044i 0.576753 1.91503i 0.430625 2.19421i 1.05512 2.44053i 0.603960 2.57589i 0.960828 + 2.66023i 0.267367 0.463092i 1.36220 + 2.85384i
19.5 −0.999687 1.00031i 1.28100 0.739583i −0.00125109 + 2.00000i −1.26649 1.84283i −2.02041 0.542044i −0.664037 2.56107i 2.00188 1.99812i −0.406034 + 0.703271i −0.577314 + 3.10913i
19.6 −0.674125 1.24320i −0.634715 + 0.366453i −1.09111 + 1.67615i −0.661137 + 2.13609i 0.883452 + 0.542044i −2.56107 + 0.664037i 2.81934 + 0.226536i −1.23142 + 2.13289i 3.10129 0.618067i
19.7 −0.444985 1.34238i −2.24836 + 1.29809i −1.60398 + 1.19468i 2.07525 0.832668i 2.74302 + 2.44053i 2.57589 + 0.603960i 2.31746 + 1.62154i 1.87009 3.23909i −2.04121 2.41525i
19.8 −0.163027 + 1.40479i −1.62820 + 0.940044i −1.94684 0.458035i −1.68493 + 1.47004i −1.05512 2.44053i 0.603960 2.57589i 0.960828 2.66023i 0.267367 0.463092i −1.79040 2.60662i
19.9 0.163027 1.40479i 1.62820 0.940044i −1.94684 0.458035i 0.430625 2.19421i −1.05512 2.44053i −0.603960 + 2.57589i −0.960828 + 2.66023i 0.267367 0.463092i −3.01219 0.962651i
19.10 0.444985 + 1.34238i 2.24836 1.29809i −1.60398 + 1.19468i 0.316513 + 2.21355i 2.74302 + 2.44053i −2.57589 0.603960i −2.31746 1.62154i 1.87009 3.23909i −2.83059 + 1.40988i
19.11 0.674125 + 1.24320i 0.634715 0.366453i −1.09111 + 1.67615i 1.51934 1.64061i 0.883452 + 0.542044i 2.56107 0.664037i −2.81934 0.226536i −1.23142 + 2.13289i 3.06384 + 0.782877i
19.12 0.999687 + 1.00031i −1.28100 + 0.739583i −0.00125109 + 2.00000i −2.22918 0.175395i −2.02041 0.542044i 0.664037 + 2.56107i −2.00188 + 1.99812i −0.406034 + 0.703271i −2.05303 2.40522i
19.13 1.13507 0.843578i 1.62820 0.940044i 0.576753 1.91503i −1.68493 + 1.47004i 1.05512 2.44053i −0.603960 + 2.57589i −0.960828 2.66023i 0.267367 0.463092i −0.672416 + 3.08996i
19.14 1.36614 + 0.365598i 1.28100 0.739583i 1.73268 + 0.998916i −2.22918 0.175395i 2.02041 0.542044i −0.664037 2.56107i 2.00188 + 1.99812i −0.406034 + 0.703271i −2.98125 1.05460i
19.15 1.38503 0.285823i −2.24836 + 1.29809i 1.83661 0.791746i 0.316513 + 2.21355i −2.74302 + 2.44053i 2.57589 + 0.603960i 2.31746 1.62154i 1.87009 3.23909i 1.07106 + 2.97537i
19.16 1.41371 0.0377920i −0.634715 + 0.366453i 1.99714 0.106854i 1.51934 1.64061i −0.883452 + 0.542044i −2.56107 + 0.664037i 2.81934 0.226536i −1.23142 + 2.13289i 2.08591 2.37676i
59.1 −1.41371 0.0377920i 0.634715 + 0.366453i 1.99714 + 0.106854i −0.661137 2.13609i −0.883452 0.542044i 2.56107 + 0.664037i −2.81934 0.226536i −1.23142 2.13289i 0.853928 + 3.04480i
59.2 −1.38503 0.285823i 2.24836 + 1.29809i 1.83661 + 0.791746i 2.07525 + 0.832668i −2.74302 2.44053i −2.57589 + 0.603960i −2.31746 1.62154i 1.87009 + 3.23909i −2.63629 1.74642i
59.3 −1.36614 + 0.365598i −1.28100 0.739583i 1.73268 0.998916i −1.26649 + 1.84283i 2.02041 + 0.542044i 0.664037 2.56107i −2.00188 + 1.99812i −0.406034 0.703271i 1.05646 2.98058i
59.4 −1.13507 0.843578i −1.62820 0.940044i 0.576753 + 1.91503i 0.430625 + 2.19421i 1.05512 + 2.44053i 0.603960 + 2.57589i 0.960828 2.66023i 0.267367 + 0.463092i 1.36220 2.85384i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.d odd 6 1 inner
20.d odd 2 1 inner
28.f even 6 1 inner
35.i odd 6 1 inner
140.s even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.s.b 32
4.b odd 2 1 inner 140.2.s.b 32
5.b even 2 1 inner 140.2.s.b 32
5.c odd 4 2 700.2.p.e 32
7.b odd 2 1 980.2.s.e 32
7.c even 3 1 980.2.c.d 32
7.c even 3 1 980.2.s.e 32
7.d odd 6 1 inner 140.2.s.b 32
7.d odd 6 1 980.2.c.d 32
20.d odd 2 1 inner 140.2.s.b 32
20.e even 4 2 700.2.p.e 32
28.d even 2 1 980.2.s.e 32
28.f even 6 1 inner 140.2.s.b 32
28.f even 6 1 980.2.c.d 32
28.g odd 6 1 980.2.c.d 32
28.g odd 6 1 980.2.s.e 32
35.c odd 2 1 980.2.s.e 32
35.i odd 6 1 inner 140.2.s.b 32
35.i odd 6 1 980.2.c.d 32
35.j even 6 1 980.2.c.d 32
35.j even 6 1 980.2.s.e 32
35.k even 12 2 700.2.p.e 32
140.c even 2 1 980.2.s.e 32
140.p odd 6 1 980.2.c.d 32
140.p odd 6 1 980.2.s.e 32
140.s even 6 1 inner 140.2.s.b 32
140.s even 6 1 980.2.c.d 32
140.x odd 12 2 700.2.p.e 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.s.b 32 1.a even 1 1 trivial
140.2.s.b 32 4.b odd 2 1 inner
140.2.s.b 32 5.b even 2 1 inner
140.2.s.b 32 7.d odd 6 1 inner
140.2.s.b 32 20.d odd 2 1 inner
140.2.s.b 32 28.f even 6 1 inner
140.2.s.b 32 35.i odd 6 1 inner
140.2.s.b 32 140.s even 6 1 inner
700.2.p.e 32 5.c odd 4 2
700.2.p.e 32 20.e even 4 2
700.2.p.e 32 35.k even 12 2
700.2.p.e 32 140.x odd 12 2
980.2.c.d 32 7.c even 3 1
980.2.c.d 32 7.d odd 6 1
980.2.c.d 32 28.f even 6 1
980.2.c.d 32 28.g odd 6 1
980.2.c.d 32 35.i odd 6 1
980.2.c.d 32 35.j even 6 1
980.2.c.d 32 140.p odd 6 1
980.2.c.d 32 140.s even 6 1
980.2.s.e 32 7.b odd 2 1
980.2.s.e 32 7.c even 3 1
980.2.s.e 32 28.d even 2 1
980.2.s.e 32 28.g odd 6 1
980.2.s.e 32 35.c odd 2 1
980.2.s.e 32 35.j even 6 1
980.2.s.e 32 140.c even 2 1
980.2.s.e 32 140.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} - 13T_{3}^{14} + 116T_{3}^{12} - 535T_{3}^{10} + 1780T_{3}^{8} - 3353T_{3}^{6} + 4445T_{3}^{4} - 2156T_{3}^{2} + 784$$ acting on $$S_{2}^{\mathrm{new}}(140, [\chi])$$.