# Properties

 Label 140.2.e.a Level $140$ Weight $2$ Character orbit 140.e 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(29,140)]

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, 1, 0]))

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

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

chi := DirichletCharacter("140.29");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + (i - 2) q^{5} - i q^{7} - 6 q^{9} +O(q^{10})$$ q + 3*i * q^3 + (i - 2) * q^5 - i * q^7 - 6 * q^9 $$q + 3 i q^{3} + (i - 2) q^{5} - i q^{7} - 6 q^{9} + 3 q^{11} + i q^{13} + ( - 6 i - 3) q^{15} + 5 i q^{17} + 8 q^{19} + 3 q^{21} + 2 i q^{23} + ( - 4 i + 3) q^{25} - 9 i q^{27} + q^{29} - 2 q^{31} + 9 i q^{33} + (2 i + 1) q^{35} - 10 i q^{37} - 3 q^{39} - 6 q^{41} - 4 i q^{43} + ( - 6 i + 12) q^{45} - 11 i q^{47} - q^{49} - 15 q^{51} + 6 i q^{53} + (3 i - 6) q^{55} + 24 i q^{57} + 10 q^{59} + 6 i q^{63} + ( - 2 i - 1) q^{65} + 10 i q^{67} - 6 q^{69} - 10 i q^{73} + (9 i + 12) q^{75} - 3 i q^{77} + 7 q^{79} + 9 q^{81} + 12 i q^{83} + ( - 10 i - 5) q^{85} + 3 i q^{87} - 8 q^{89} + q^{91} - 6 i q^{93} + (8 i - 16) q^{95} - 3 i q^{97} - 18 q^{99} +O(q^{100})$$ q + 3*i * q^3 + (i - 2) * q^5 - i * q^7 - 6 * q^9 + 3 * q^11 + i * q^13 + (-6*i - 3) * q^15 + 5*i * q^17 + 8 * q^19 + 3 * q^21 + 2*i * q^23 + (-4*i + 3) * q^25 - 9*i * q^27 + q^29 - 2 * q^31 + 9*i * q^33 + (2*i + 1) * q^35 - 10*i * q^37 - 3 * q^39 - 6 * q^41 - 4*i * q^43 + (-6*i + 12) * q^45 - 11*i * q^47 - q^49 - 15 * q^51 + 6*i * q^53 + (3*i - 6) * q^55 + 24*i * q^57 + 10 * q^59 + 6*i * q^63 + (-2*i - 1) * q^65 + 10*i * q^67 - 6 * q^69 - 10*i * q^73 + (9*i + 12) * q^75 - 3*i * q^77 + 7 * q^79 + 9 * q^81 + 12*i * q^83 + (-10*i - 5) * q^85 + 3*i * q^87 - 8 * q^89 + q^91 - 6*i * q^93 + (8*i - 16) * q^95 - 3*i * q^97 - 18 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5} - 12 q^{9}+O(q^{10})$$ 2 * q - 4 * q^5 - 12 * q^9 $$2 q - 4 q^{5} - 12 q^{9} + 6 q^{11} - 6 q^{15} + 16 q^{19} + 6 q^{21} + 6 q^{25} + 2 q^{29} - 4 q^{31} + 2 q^{35} - 6 q^{39} - 12 q^{41} + 24 q^{45} - 2 q^{49} - 30 q^{51} - 12 q^{55} + 20 q^{59} - 2 q^{65} - 12 q^{69} + 24 q^{75} + 14 q^{79} + 18 q^{81} - 10 q^{85} - 16 q^{89} + 2 q^{91} - 32 q^{95} - 36 q^{99}+O(q^{100})$$ 2 * q - 4 * q^5 - 12 * q^9 + 6 * q^11 - 6 * q^15 + 16 * q^19 + 6 * q^21 + 6 * q^25 + 2 * q^29 - 4 * q^31 + 2 * q^35 - 6 * q^39 - 12 * q^41 + 24 * q^45 - 2 * q^49 - 30 * q^51 - 12 * q^55 + 20 * q^59 - 2 * q^65 - 12 * q^69 + 24 * q^75 + 14 * q^79 + 18 * q^81 - 10 * q^85 - 16 * q^89 + 2 * q^91 - 32 * q^95 - 36 * q^99

## Character values

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

 $$n$$ $$57$$ $$71$$ $$101$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 − 1.00000i 1.00000i
0 3.00000i 0 −2.00000 1.00000i 0 1.00000i 0 −6.00000 0
29.2 0 3.00000i 0 −2.00000 + 1.00000i 0 1.00000i 0 −6.00000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.e.a 2
3.b odd 2 1 1260.2.k.c 2
4.b odd 2 1 560.2.g.a 2
5.b even 2 1 inner 140.2.e.a 2
5.c odd 4 1 700.2.a.a 1
5.c odd 4 1 700.2.a.j 1
7.b odd 2 1 980.2.e.b 2
7.c even 3 2 980.2.q.f 4
7.d odd 6 2 980.2.q.c 4
8.b even 2 1 2240.2.g.e 2
8.d odd 2 1 2240.2.g.f 2
12.b even 2 1 5040.2.t.s 2
15.d odd 2 1 1260.2.k.c 2
15.e even 4 1 6300.2.a.c 1
15.e even 4 1 6300.2.a.t 1
20.d odd 2 1 560.2.g.a 2
20.e even 4 1 2800.2.a.a 1
20.e even 4 1 2800.2.a.bf 1
35.c odd 2 1 980.2.e.b 2
35.f even 4 1 4900.2.a.b 1
35.f even 4 1 4900.2.a.w 1
35.i odd 6 2 980.2.q.c 4
35.j even 6 2 980.2.q.f 4
40.e odd 2 1 2240.2.g.f 2
40.f even 2 1 2240.2.g.e 2
60.h even 2 1 5040.2.t.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 1.a even 1 1 trivial
140.2.e.a 2 5.b even 2 1 inner
560.2.g.a 2 4.b odd 2 1
560.2.g.a 2 20.d odd 2 1
700.2.a.a 1 5.c odd 4 1
700.2.a.j 1 5.c odd 4 1
980.2.e.b 2 7.b odd 2 1
980.2.e.b 2 35.c odd 2 1
980.2.q.c 4 7.d odd 6 2
980.2.q.c 4 35.i odd 6 2
980.2.q.f 4 7.c even 3 2
980.2.q.f 4 35.j even 6 2
1260.2.k.c 2 3.b odd 2 1
1260.2.k.c 2 15.d odd 2 1
2240.2.g.e 2 8.b even 2 1
2240.2.g.e 2 40.f even 2 1
2240.2.g.f 2 8.d odd 2 1
2240.2.g.f 2 40.e odd 2 1
2800.2.a.a 1 20.e even 4 1
2800.2.a.bf 1 20.e even 4 1
4900.2.a.b 1 35.f even 4 1
4900.2.a.w 1 35.f even 4 1
5040.2.t.s 2 12.b even 2 1
5040.2.t.s 2 60.h even 2 1
6300.2.a.c 1 15.e even 4 1
6300.2.a.t 1 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(140, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 25$$
$19$ $$(T - 8)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 121$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 10)^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 100$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 7)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 8)^{2}$$
$97$ $$T^{2} + 9$$