# Properties

 Label 140.2.q.b Level $140$ Weight $2$ Character orbit 140.q Analytic conductor $1.118$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.11790562830$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - \beta_{2} ) q^{3} + ( -1 + \beta_{1} - \beta_{3} ) q^{5} + ( 1 + \beta_{1} - \beta_{2} ) q^{7} +O(q^{10})$$ $$q + ( 2 - \beta_{2} ) q^{3} + ( -1 + \beta_{1} - \beta_{3} ) q^{5} + ( 1 + \beta_{1} - \beta_{2} ) q^{7} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{13} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{15} + ( -3 + \beta_{2} - \beta_{3} ) q^{17} + ( -1 + \beta_{1} - 2 \beta_{3} ) q^{19} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{21} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{23} + ( -4 - \beta_{1} + \beta_{3} ) q^{25} + ( -3 + 6 \beta_{2} ) q^{27} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{29} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + ( 3 - 3 \beta_{1} ) q^{33} + ( -6 + \beta_{1} + 6 \beta_{2} ) q^{35} + ( -5 + \beta_{1} - 4 \beta_{2} ) q^{37} + ( 2 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{39} + ( -7 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{41} + ( 3 - \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{43} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{47} + ( 4 \beta_{2} + 3 \beta_{3} ) q^{49} + ( -5 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{51} + ( 1 + \beta_{2} + 3 \beta_{3} ) q^{53} + ( 9 + \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{55} + ( -2 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{57} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{59} + ( 5 + 2 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} ) q^{61} + ( 8 + 2 \beta_{1} + 2 \beta_{2} ) q^{65} + ( 6 - \beta_{2} + 4 \beta_{3} ) q^{67} + ( 7 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{69} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 5 - 3 \beta_{2} - \beta_{3} ) q^{73} + ( -8 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{75} + ( 7 - 11 \beta_{2} - 3 \beta_{3} ) q^{77} + ( 3 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{79} + 9 \beta_{2} q^{81} + ( -1 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{83} + ( 3 - 3 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{85} + ( -1 + 2 \beta_{2} + 3 \beta_{3} ) q^{87} + ( -7 + 7 \beta_{2} ) q^{89} + ( 10 - 4 \beta_{1} - 12 \beta_{2} + 2 \beta_{3} ) q^{91} + ( -1 + 3 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -4 - \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{95} + ( -4 + 8 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + O(q^{10})$$ $$4 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{11} - 3 q^{15} - 9 q^{17} - q^{19} + 12 q^{23} - 18 q^{25} - 2 q^{29} + q^{31} + 9 q^{33} - 11 q^{35} - 27 q^{37} - 6 q^{39} - 30 q^{41} + 15 q^{47} + 5 q^{49} - 9 q^{51} + 3 q^{53} + 27 q^{55} + q^{59} + 12 q^{61} + 38 q^{65} + 18 q^{67} + 24 q^{69} + 12 q^{71} + 15 q^{73} - 27 q^{75} + 9 q^{77} + 7 q^{79} + 18 q^{81} - 5 q^{85} - 3 q^{87} - 14 q^{89} + 10 q^{91} + 3 q^{93} - 28 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 5$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu + 5$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{3} + 4 \beta_{1} + 5$$

## Character values

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

 $$n$$ $$57$$ $$71$$ $$101$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1 + \beta_{2}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 −1.63746 − 1.52274i 2.13746 + 0.656712i 2.13746 − 0.656712i −1.63746 + 1.52274i
0 1.50000 0.866025i 0 −0.500000 2.17945i 0 −1.13746 2.38876i 0 0 0
9.2 0 1.50000 0.866025i 0 −0.500000 + 2.17945i 0 2.63746 0.209313i 0 0 0
109.1 0 1.50000 + 0.866025i 0 −0.500000 2.17945i 0 2.63746 + 0.209313i 0 0 0
109.2 0 1.50000 + 0.866025i 0 −0.500000 + 2.17945i 0 −1.13746 + 2.38876i 0 0 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
35.j 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.q.b yes 4
3.b odd 2 1 1260.2.bm.b 4
4.b odd 2 1 560.2.bw.a 4
5.b even 2 1 140.2.q.a 4
5.c odd 4 2 700.2.i.f 8
7.b odd 2 1 980.2.q.b 4
7.c even 3 1 140.2.q.a 4
7.c even 3 1 980.2.e.f 4
7.d odd 6 1 980.2.e.c 4
7.d odd 6 1 980.2.q.g 4
15.d odd 2 1 1260.2.bm.a 4
20.d odd 2 1 560.2.bw.e 4
21.h odd 6 1 1260.2.bm.a 4
28.g odd 6 1 560.2.bw.e 4
35.c odd 2 1 980.2.q.g 4
35.i odd 6 1 980.2.e.c 4
35.i odd 6 1 980.2.q.b 4
35.j even 6 1 inner 140.2.q.b yes 4
35.j even 6 1 980.2.e.f 4
35.k even 12 2 4900.2.a.bf 4
35.l odd 12 2 700.2.i.f 8
35.l odd 12 2 4900.2.a.be 4
105.o odd 6 1 1260.2.bm.b 4
140.p odd 6 1 560.2.bw.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.q.a 4 5.b even 2 1
140.2.q.a 4 7.c even 3 1
140.2.q.b yes 4 1.a even 1 1 trivial
140.2.q.b yes 4 35.j even 6 1 inner
560.2.bw.a 4 4.b odd 2 1
560.2.bw.a 4 140.p odd 6 1
560.2.bw.e 4 20.d odd 2 1
560.2.bw.e 4 28.g odd 6 1
700.2.i.f 8 5.c odd 4 2
700.2.i.f 8 35.l odd 12 2
980.2.e.c 4 7.d odd 6 1
980.2.e.c 4 35.i odd 6 1
980.2.e.f 4 7.c even 3 1
980.2.e.f 4 35.j even 6 1
980.2.q.b 4 7.b odd 2 1
980.2.q.b 4 35.i odd 6 1
980.2.q.g 4 7.d odd 6 1
980.2.q.g 4 35.c odd 2 1
1260.2.bm.a 4 15.d odd 2 1
1260.2.bm.a 4 21.h odd 6 1
1260.2.bm.b 4 3.b odd 2 1
1260.2.bm.b 4 105.o odd 6 1
4900.2.a.be 4 35.l odd 12 2
4900.2.a.bf 4 35.k even 12 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 - 3 T + T^{2} )^{2}$$
$5$ $$( 5 + T + T^{2} )^{2}$$
$7$ $$49 - 21 T + 2 T^{2} - 3 T^{3} + T^{4}$$
$11$ $$144 + 36 T + 21 T^{2} - 3 T^{3} + T^{4}$$
$13$ $$256 + 44 T^{2} + T^{4}$$
$17$ $$4 + 18 T + 29 T^{2} + 9 T^{3} + T^{4}$$
$19$ $$196 - 14 T + 15 T^{2} + T^{3} + T^{4}$$
$23$ $$49 + 84 T + 41 T^{2} - 12 T^{3} + T^{4}$$
$29$ $$( -14 + T + T^{2} )^{2}$$
$31$ $$196 + 14 T + 15 T^{2} - T^{3} + T^{4}$$
$37$ $$3136 + 1512 T + 299 T^{2} + 27 T^{3} + T^{4}$$
$41$ $$( 42 + 15 T + T^{2} )^{2}$$
$43$ $$196 + 47 T^{2} + T^{4}$$
$47$ $$196 - 210 T + 89 T^{2} - 15 T^{3} + T^{4}$$
$53$ $$1764 + 126 T - 39 T^{2} - 3 T^{3} + T^{4}$$
$59$ $$196 + 14 T + 15 T^{2} - T^{3} + T^{4}$$
$61$ $$441 + 252 T + 165 T^{2} - 12 T^{3} + T^{4}$$
$67$ $$2401 + 882 T + 59 T^{2} - 18 T^{3} + T^{4}$$
$71$ $$( -48 - 6 T + T^{2} )^{2}$$
$73$ $$196 - 210 T + 89 T^{2} - 15 T^{3} + T^{4}$$
$79$ $$4 + 14 T + 51 T^{2} - 7 T^{3} + T^{4}$$
$83$ $$1764 + 87 T^{2} + T^{4}$$
$89$ $$( 49 + 7 T + T^{2} )^{2}$$
$97$ $$( 48 + T^{2} )^{2}$$