# Properties

 Label 140.2.i.a Level $140$ Weight $2$ Character orbit 140.i Analytic conductor $1.118$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,2,Mod(81,140)]

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([0, 0, 4]))

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

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

chi := DirichletCharacter("140.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$140 = 2^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 140.i (of order $$3$$, 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} + \zeta_{6} q^{5} + ( - \zeta_{6} + 3) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + (z - 1) * q^3 + z * q^5 + (-z + 3) * q^7 + 2*z * q^9 $$q + (\zeta_{6} - 1) q^{3} + \zeta_{6} q^{5} + ( - \zeta_{6} + 3) q^{7} + 2 \zeta_{6} q^{9} + (6 \zeta_{6} - 6) q^{11} + 2 q^{13} - q^{15} + ( - 6 \zeta_{6} + 6) q^{17} - 8 \zeta_{6} q^{19} + (3 \zeta_{6} - 2) q^{21} - 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 5 q^{27} + 3 q^{29} + (2 \zeta_{6} - 2) q^{31} - 6 \zeta_{6} q^{33} + (2 \zeta_{6} + 1) q^{35} - 8 \zeta_{6} q^{37} + (2 \zeta_{6} - 2) q^{39} - 3 q^{41} + 5 q^{43} + (2 \zeta_{6} - 2) q^{45} + ( - 5 \zeta_{6} + 8) q^{49} + 6 \zeta_{6} q^{51} + (12 \zeta_{6} - 12) q^{53} - 6 q^{55} + 8 q^{57} + \zeta_{6} q^{61} + (4 \zeta_{6} + 2) q^{63} + 2 \zeta_{6} q^{65} + ( - 7 \zeta_{6} + 7) q^{67} + 3 q^{69} + ( - 10 \zeta_{6} + 10) q^{73} - \zeta_{6} q^{75} + (18 \zeta_{6} - 12) q^{77} + 4 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 3 q^{83} + 6 q^{85} + (3 \zeta_{6} - 3) q^{87} + 3 \zeta_{6} q^{89} + ( - 2 \zeta_{6} + 6) q^{91} - 2 \zeta_{6} q^{93} + ( - 8 \zeta_{6} + 8) q^{95} - 10 q^{97} - 12 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 + z * q^5 + (-z + 3) * q^7 + 2*z * q^9 + (6*z - 6) * q^11 + 2 * q^13 - q^15 + (-6*z + 6) * q^17 - 8*z * q^19 + (3*z - 2) * q^21 - 3*z * q^23 + (z - 1) * q^25 - 5 * q^27 + 3 * q^29 + (2*z - 2) * q^31 - 6*z * q^33 + (2*z + 1) * q^35 - 8*z * q^37 + (2*z - 2) * q^39 - 3 * q^41 + 5 * q^43 + (2*z - 2) * q^45 + (-5*z + 8) * q^49 + 6*z * q^51 + (12*z - 12) * q^53 - 6 * q^55 + 8 * q^57 + z * q^61 + (4*z + 2) * q^63 + 2*z * q^65 + (-7*z + 7) * q^67 + 3 * q^69 + (-10*z + 10) * q^73 - z * q^75 + (18*z - 12) * q^77 + 4*z * q^79 + (z - 1) * q^81 + 3 * q^83 + 6 * q^85 + (3*z - 3) * q^87 + 3*z * q^89 + (-2*z + 6) * q^91 - 2*z * q^93 + (-8*z + 8) * q^95 - 10 * q^97 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - q^3 + q^5 + 5 * q^7 + 2 * q^9 $$2 q - q^{3} + q^{5} + 5 q^{7} + 2 q^{9} - 6 q^{11} + 4 q^{13} - 2 q^{15} + 6 q^{17} - 8 q^{19} - q^{21} - 3 q^{23} - q^{25} - 10 q^{27} + 6 q^{29} - 2 q^{31} - 6 q^{33} + 4 q^{35} - 8 q^{37} - 2 q^{39} - 6 q^{41} + 10 q^{43} - 2 q^{45} + 11 q^{49} + 6 q^{51} - 12 q^{53} - 12 q^{55} + 16 q^{57} + q^{61} + 8 q^{63} + 2 q^{65} + 7 q^{67} + 6 q^{69} + 10 q^{73} - q^{75} - 6 q^{77} + 4 q^{79} - q^{81} + 6 q^{83} + 12 q^{85} - 3 q^{87} + 3 q^{89} + 10 q^{91} - 2 q^{93} + 8 q^{95} - 20 q^{97} - 24 q^{99}+O(q^{100})$$ 2 * q - q^3 + q^5 + 5 * q^7 + 2 * q^9 - 6 * q^11 + 4 * q^13 - 2 * q^15 + 6 * q^17 - 8 * q^19 - q^21 - 3 * q^23 - q^25 - 10 * q^27 + 6 * q^29 - 2 * q^31 - 6 * q^33 + 4 * q^35 - 8 * q^37 - 2 * q^39 - 6 * q^41 + 10 * q^43 - 2 * q^45 + 11 * q^49 + 6 * q^51 - 12 * q^53 - 12 * q^55 + 16 * q^57 + q^61 + 8 * q^63 + 2 * q^65 + 7 * q^67 + 6 * q^69 + 10 * q^73 - q^75 - 6 * q^77 + 4 * q^79 - q^81 + 6 * q^83 + 12 * q^85 - 3 * q^87 + 3 * q^89 + 10 * q^91 - 2 * q^93 + 8 * q^95 - 20 * q^97 - 24 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/140\mathbb{Z}\right)^\times$$.

 $$n$$ $$57$$ $$71$$ $$101$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 2.50000 0.866025i 0 1.00000 + 1.73205i 0
121.1 0 −0.500000 0.866025i 0 0.500000 0.866025i 0 2.50000 + 0.866025i 0 1.00000 1.73205i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.i.a 2
3.b odd 2 1 1260.2.s.c 2
4.b odd 2 1 560.2.q.f 2
5.b even 2 1 700.2.i.b 2
5.c odd 4 2 700.2.r.a 4
7.b odd 2 1 980.2.i.f 2
7.c even 3 1 inner 140.2.i.a 2
7.c even 3 1 980.2.a.g 1
7.d odd 6 1 980.2.a.e 1
7.d odd 6 1 980.2.i.f 2
21.g even 6 1 8820.2.a.a 1
21.h odd 6 1 1260.2.s.c 2
21.h odd 6 1 8820.2.a.p 1
28.f even 6 1 3920.2.a.w 1
28.g odd 6 1 560.2.q.f 2
28.g odd 6 1 3920.2.a.k 1
35.i odd 6 1 4900.2.a.q 1
35.j even 6 1 700.2.i.b 2
35.j even 6 1 4900.2.a.i 1
35.k even 12 2 4900.2.e.n 2
35.l odd 12 2 700.2.r.a 4
35.l odd 12 2 4900.2.e.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.a 2 1.a even 1 1 trivial
140.2.i.a 2 7.c even 3 1 inner
560.2.q.f 2 4.b odd 2 1
560.2.q.f 2 28.g odd 6 1
700.2.i.b 2 5.b even 2 1
700.2.i.b 2 35.j even 6 1
700.2.r.a 4 5.c odd 4 2
700.2.r.a 4 35.l odd 12 2
980.2.a.e 1 7.d odd 6 1
980.2.a.g 1 7.c even 3 1
980.2.i.f 2 7.b odd 2 1
980.2.i.f 2 7.d odd 6 1
1260.2.s.c 2 3.b odd 2 1
1260.2.s.c 2 21.h odd 6 1
3920.2.a.k 1 28.g odd 6 1
3920.2.a.w 1 28.f even 6 1
4900.2.a.i 1 35.j even 6 1
4900.2.a.q 1 35.i odd 6 1
4900.2.e.m 2 35.l odd 12 2
4900.2.e.n 2 35.k even 12 2
8820.2.a.a 1 21.g even 6 1
8820.2.a.p 1 21.h odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} - 5T + 7$$
$11$ $$T^{2} + 6T + 36$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 8T + 64$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2} + 2T + 4$$
$37$ $$T^{2} + 8T + 64$$
$41$ $$(T + 3)^{2}$$
$43$ $$(T - 5)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 12T + 144$$
$59$ $$T^{2}$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} - 7T + 49$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 10T + 100$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$(T - 3)^{2}$$
$89$ $$T^{2} - 3T + 9$$
$97$ $$(T + 10)^{2}$$