# Properties

 Label 140.2.i.b Level $140$ Weight $2$ Character orbit 140.i Analytic conductor $1.118$ Analytic rank $0$ Dimension $2$ 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

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

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + ( - 3 \zeta_{6} + 3) q^{3} + \zeta_{6} q^{5} + (3 \zeta_{6} - 1) q^{7} - 6 \zeta_{6} q^{9} +O(q^{10})$$ q + (-3*z + 3) * q^3 + z * q^5 + (3*z - 1) * q^7 - 6*z * q^9 $$q + ( - 3 \zeta_{6} + 3) q^{3} + \zeta_{6} q^{5} + (3 \zeta_{6} - 1) q^{7} - 6 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} - 6 q^{13} + 3 q^{15} + (2 \zeta_{6} - 2) q^{17} + (3 \zeta_{6} + 6) q^{21} + 9 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 9 q^{27} + 3 q^{29} + (2 \zeta_{6} - 2) q^{31} - 6 \zeta_{6} q^{33} + (2 \zeta_{6} - 3) q^{35} - 8 \zeta_{6} q^{37} + (18 \zeta_{6} - 18) q^{39} + 5 q^{41} + q^{43} + ( - 6 \zeta_{6} + 6) q^{45} - 8 \zeta_{6} q^{47} + (3 \zeta_{6} - 8) q^{49} + 6 \zeta_{6} q^{51} + (4 \zeta_{6} - 4) q^{53} + 2 q^{55} + ( - 8 \zeta_{6} + 8) q^{59} - 7 \zeta_{6} q^{61} + ( - 12 \zeta_{6} + 18) q^{63} - 6 \zeta_{6} q^{65} + ( - 3 \zeta_{6} + 3) q^{67} + 27 q^{69} + 8 q^{71} + (14 \zeta_{6} - 14) q^{73} + 3 \zeta_{6} q^{75} + (2 \zeta_{6} + 4) q^{77} - 4 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} - q^{83} - 2 q^{85} + ( - 9 \zeta_{6} + 9) q^{87} - 13 \zeta_{6} q^{89} + ( - 18 \zeta_{6} + 6) q^{91} + 6 \zeta_{6} q^{93} - 10 q^{97} - 12 q^{99} +O(q^{100})$$ q + (-3*z + 3) * q^3 + z * q^5 + (3*z - 1) * q^7 - 6*z * q^9 + (-2*z + 2) * q^11 - 6 * q^13 + 3 * q^15 + (2*z - 2) * q^17 + (3*z + 6) * q^21 + 9*z * q^23 + (z - 1) * q^25 - 9 * q^27 + 3 * q^29 + (2*z - 2) * q^31 - 6*z * q^33 + (2*z - 3) * q^35 - 8*z * q^37 + (18*z - 18) * q^39 + 5 * q^41 + q^43 + (-6*z + 6) * q^45 - 8*z * q^47 + (3*z - 8) * q^49 + 6*z * q^51 + (4*z - 4) * q^53 + 2 * q^55 + (-8*z + 8) * q^59 - 7*z * q^61 + (-12*z + 18) * q^63 - 6*z * q^65 + (-3*z + 3) * q^67 + 27 * q^69 + 8 * q^71 + (14*z - 14) * q^73 + 3*z * q^75 + (2*z + 4) * q^77 - 4*z * q^79 + (9*z - 9) * q^81 - q^83 - 2 * q^85 + (-9*z + 9) * q^87 - 13*z * q^89 + (-18*z + 6) * q^91 + 6*z * q^93 - 10 * q^97 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + q^{5} + q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + q^5 + q^7 - 6 * q^9 $$2 q + 3 q^{3} + q^{5} + q^{7} - 6 q^{9} + 2 q^{11} - 12 q^{13} + 6 q^{15} - 2 q^{17} + 15 q^{21} + 9 q^{23} - q^{25} - 18 q^{27} + 6 q^{29} - 2 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} - 18 q^{39} + 10 q^{41} + 2 q^{43} + 6 q^{45} - 8 q^{47} - 13 q^{49} + 6 q^{51} - 4 q^{53} + 4 q^{55} + 8 q^{59} - 7 q^{61} + 24 q^{63} - 6 q^{65} + 3 q^{67} + 54 q^{69} + 16 q^{71} - 14 q^{73} + 3 q^{75} + 10 q^{77} - 4 q^{79} - 9 q^{81} - 2 q^{83} - 4 q^{85} + 9 q^{87} - 13 q^{89} - 6 q^{91} + 6 q^{93} - 20 q^{97} - 24 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 + q^5 + q^7 - 6 * q^9 + 2 * q^11 - 12 * q^13 + 6 * q^15 - 2 * q^17 + 15 * q^21 + 9 * q^23 - q^25 - 18 * q^27 + 6 * q^29 - 2 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 - 18 * q^39 + 10 * q^41 + 2 * q^43 + 6 * q^45 - 8 * q^47 - 13 * q^49 + 6 * q^51 - 4 * q^53 + 4 * q^55 + 8 * q^59 - 7 * q^61 + 24 * q^63 - 6 * q^65 + 3 * q^67 + 54 * q^69 + 16 * q^71 - 14 * q^73 + 3 * q^75 + 10 * q^77 - 4 * q^79 - 9 * q^81 - 2 * q^83 - 4 * q^85 + 9 * q^87 - 13 * q^89 - 6 * q^91 + 6 * q^93 - 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.

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 1.50000 2.59808i 0 0.500000 + 0.866025i 0 0.500000 + 2.59808i 0 −3.00000 5.19615i 0
121.1 0 1.50000 + 2.59808i 0 0.500000 0.866025i 0 0.500000 2.59808i 0 −3.00000 + 5.19615i 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.b 2
3.b odd 2 1 1260.2.s.b 2
4.b odd 2 1 560.2.q.a 2
5.b even 2 1 700.2.i.a 2
5.c odd 4 2 700.2.r.c 4
7.b odd 2 1 980.2.i.a 2
7.c even 3 1 inner 140.2.i.b 2
7.c even 3 1 980.2.a.a 1
7.d odd 6 1 980.2.a.i 1
7.d odd 6 1 980.2.i.a 2
21.g even 6 1 8820.2.a.k 1
21.h odd 6 1 1260.2.s.b 2
21.h odd 6 1 8820.2.a.w 1
28.f even 6 1 3920.2.a.d 1
28.g odd 6 1 560.2.q.a 2
28.g odd 6 1 3920.2.a.bi 1
35.i odd 6 1 4900.2.a.a 1
35.j even 6 1 700.2.i.a 2
35.j even 6 1 4900.2.a.v 1
35.k even 12 2 4900.2.e.b 2
35.l odd 12 2 700.2.r.c 4
35.l odd 12 2 4900.2.e.c 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 9$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} - T + 7$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$(T + 6)^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 9T + 81$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2} + 2T + 4$$
$37$ $$T^{2} + 8T + 64$$
$41$ $$(T - 5)^{2}$$
$43$ $$(T - 1)^{2}$$
$47$ $$T^{2} + 8T + 64$$
$53$ $$T^{2} + 4T + 16$$
$59$ $$T^{2} - 8T + 64$$
$61$ $$T^{2} + 7T + 49$$
$67$ $$T^{2} - 3T + 9$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} + 14T + 196$$
$79$ $$T^{2} + 4T + 16$$
$83$ $$(T + 1)^{2}$$
$89$ $$T^{2} + 13T + 169$$
$97$ $$(T + 10)^{2}$$