# Properties

 Label 70.2.e.b Level $70$ Weight $2$ Character orbit 70.e Analytic conductor $0.559$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$70 = 2 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 70.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.558952814149$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( 2 - 2 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} -2 q^{6} + ( -1 - 2 \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( 2 - 2 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} -2 q^{6} + ( -1 - 2 \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{10} + ( -3 + 3 \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{12} + 5 q^{13} + ( -2 + 3 \zeta_{6} ) q^{14} + 2 q^{15} -\zeta_{6} q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + ( -1 + \zeta_{6} ) q^{18} + \zeta_{6} q^{19} - q^{20} + ( -6 + 2 \zeta_{6} ) q^{21} + 3 q^{22} -3 \zeta_{6} q^{23} + ( 2 - 2 \zeta_{6} ) q^{24} + ( -1 + \zeta_{6} ) q^{25} -5 \zeta_{6} q^{26} + 4 q^{27} + ( 3 - \zeta_{6} ) q^{28} -6 q^{29} -2 \zeta_{6} q^{30} + ( 4 - 4 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + 6 \zeta_{6} q^{33} + 6 q^{34} + ( 2 - 3 \zeta_{6} ) q^{35} + q^{36} -11 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{38} + ( 10 - 10 \zeta_{6} ) q^{39} + \zeta_{6} q^{40} + 3 q^{41} + ( 2 + 4 \zeta_{6} ) q^{42} -10 q^{43} -3 \zeta_{6} q^{44} + ( 1 - \zeta_{6} ) q^{45} + ( -3 + 3 \zeta_{6} ) q^{46} -3 \zeta_{6} q^{47} -2 q^{48} + ( -3 + 8 \zeta_{6} ) q^{49} + q^{50} + 12 \zeta_{6} q^{51} + ( -5 + 5 \zeta_{6} ) q^{52} + ( -3 + 3 \zeta_{6} ) q^{53} -4 \zeta_{6} q^{54} -3 q^{55} + ( -1 - 2 \zeta_{6} ) q^{56} + 2 q^{57} + 6 \zeta_{6} q^{58} + ( -2 + 2 \zeta_{6} ) q^{60} + 4 \zeta_{6} q^{61} -4 q^{62} + ( -2 + 3 \zeta_{6} ) q^{63} + q^{64} + 5 \zeta_{6} q^{65} + ( 6 - 6 \zeta_{6} ) q^{66} + ( 4 - 4 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{68} -6 q^{69} + ( -3 + \zeta_{6} ) q^{70} + 12 q^{71} -\zeta_{6} q^{72} + ( 4 - 4 \zeta_{6} ) q^{73} + ( -11 + 11 \zeta_{6} ) q^{74} + 2 \zeta_{6} q^{75} - q^{76} + ( 9 - 3 \zeta_{6} ) q^{77} -10 q^{78} + 10 \zeta_{6} q^{79} + ( 1 - \zeta_{6} ) q^{80} + ( 11 - 11 \zeta_{6} ) q^{81} -3 \zeta_{6} q^{82} -12 q^{83} + ( 4 - 6 \zeta_{6} ) q^{84} -6 q^{85} + 10 \zeta_{6} q^{86} + ( -12 + 12 \zeta_{6} ) q^{87} + ( -3 + 3 \zeta_{6} ) q^{88} -6 \zeta_{6} q^{89} - q^{90} + ( -5 - 10 \zeta_{6} ) q^{91} + 3 q^{92} -8 \zeta_{6} q^{93} + ( -3 + 3 \zeta_{6} ) q^{94} + ( -1 + \zeta_{6} ) q^{95} + 2 \zeta_{6} q^{96} + 14 q^{97} + ( 8 - 5 \zeta_{6} ) q^{98} + 3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 2q^{3} - q^{4} + q^{5} - 4q^{6} - 4q^{7} + 2q^{8} - q^{9} + O(q^{10})$$ $$2q - q^{2} + 2q^{3} - q^{4} + q^{5} - 4q^{6} - 4q^{7} + 2q^{8} - q^{9} + q^{10} - 3q^{11} + 2q^{12} + 10q^{13} - q^{14} + 4q^{15} - q^{16} - 6q^{17} - q^{18} + q^{19} - 2q^{20} - 10q^{21} + 6q^{22} - 3q^{23} + 2q^{24} - q^{25} - 5q^{26} + 8q^{27} + 5q^{28} - 12q^{29} - 2q^{30} + 4q^{31} - q^{32} + 6q^{33} + 12q^{34} + q^{35} + 2q^{36} - 11q^{37} + q^{38} + 10q^{39} + q^{40} + 6q^{41} + 8q^{42} - 20q^{43} - 3q^{44} + q^{45} - 3q^{46} - 3q^{47} - 4q^{48} + 2q^{49} + 2q^{50} + 12q^{51} - 5q^{52} - 3q^{53} - 4q^{54} - 6q^{55} - 4q^{56} + 4q^{57} + 6q^{58} - 2q^{60} + 4q^{61} - 8q^{62} - q^{63} + 2q^{64} + 5q^{65} + 6q^{66} + 4q^{67} - 6q^{68} - 12q^{69} - 5q^{70} + 24q^{71} - q^{72} + 4q^{73} - 11q^{74} + 2q^{75} - 2q^{76} + 15q^{77} - 20q^{78} + 10q^{79} + q^{80} + 11q^{81} - 3q^{82} - 24q^{83} + 2q^{84} - 12q^{85} + 10q^{86} - 12q^{87} - 3q^{88} - 6q^{89} - 2q^{90} - 20q^{91} + 6q^{92} - 8q^{93} - 3q^{94} - q^{95} + 2q^{96} + 28q^{97} + 11q^{98} + 6q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/70\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$57$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 0.866025i 1.00000 1.73205i −0.500000 + 0.866025i 0.500000 + 0.866025i −2.00000 −2.00000 1.73205i 1.00000 −0.500000 0.866025i 0.500000 0.866025i
51.1 −0.500000 + 0.866025i 1.00000 + 1.73205i −0.500000 0.866025i 0.500000 0.866025i −2.00000 −2.00000 + 1.73205i 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
 $$n$$: 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 70.2.e.b 2
3.b odd 2 1 630.2.k.e 2
4.b odd 2 1 560.2.q.d 2
5.b even 2 1 350.2.e.h 2
5.c odd 4 2 350.2.j.a 4
7.b odd 2 1 490.2.e.a 2
7.c even 3 1 inner 70.2.e.b 2
7.c even 3 1 490.2.a.g 1
7.d odd 6 1 490.2.a.j 1
7.d odd 6 1 490.2.e.a 2
21.g even 6 1 4410.2.a.c 1
21.h odd 6 1 630.2.k.e 2
21.h odd 6 1 4410.2.a.m 1
28.f even 6 1 3920.2.a.g 1
28.g odd 6 1 560.2.q.d 2
28.g odd 6 1 3920.2.a.be 1
35.i odd 6 1 2450.2.a.f 1
35.j even 6 1 350.2.e.h 2
35.j even 6 1 2450.2.a.p 1
35.k even 12 2 2450.2.c.p 2
35.l odd 12 2 350.2.j.a 4
35.l odd 12 2 2450.2.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 1.a even 1 1 trivial
70.2.e.b 2 7.c even 3 1 inner
350.2.e.h 2 5.b even 2 1
350.2.e.h 2 35.j even 6 1
350.2.j.a 4 5.c odd 4 2
350.2.j.a 4 35.l odd 12 2
490.2.a.g 1 7.c even 3 1
490.2.a.j 1 7.d odd 6 1
490.2.e.a 2 7.b odd 2 1
490.2.e.a 2 7.d odd 6 1
560.2.q.d 2 4.b odd 2 1
560.2.q.d 2 28.g odd 6 1
630.2.k.e 2 3.b odd 2 1
630.2.k.e 2 21.h odd 6 1
2450.2.a.f 1 35.i odd 6 1
2450.2.a.p 1 35.j even 6 1
2450.2.c.f 2 35.l odd 12 2
2450.2.c.p 2 35.k even 12 2
3920.2.a.g 1 28.f even 6 1
3920.2.a.be 1 28.g odd 6 1
4410.2.a.c 1 21.g even 6 1
4410.2.a.m 1 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(70, [\chi])$$:

 $$T_{3}^{2} - 2 T_{3} + 4$$ $$T_{13} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$4 - 2 T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$( -5 + T )^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$121 + 11 T + T^{2}$$
$41$ $$( -3 + T )^{2}$$
$43$ $$( 10 + T )^{2}$$
$47$ $$9 + 3 T + T^{2}$$
$53$ $$9 + 3 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$16 - 4 T + T^{2}$$
$67$ $$16 - 4 T + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$16 - 4 T + T^{2}$$
$79$ $$100 - 10 T + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( -14 + T )^{2}$$