# Properties

 Label 70.2.e.c 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} - q^{6} + ( 1 - 3 \zeta_{6} ) q^{7} - q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} - q^{6} + ( 1 - 3 \zeta_{6} ) q^{7} - q^{8} + 2 \zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{10} + ( 6 - 6 \zeta_{6} ) q^{11} -\zeta_{6} q^{12} -4 q^{13} + ( 3 - 2 \zeta_{6} ) q^{14} - q^{15} -\zeta_{6} q^{16} + ( -2 + 2 \zeta_{6} ) q^{18} -2 \zeta_{6} q^{19} - q^{20} + ( 2 + \zeta_{6} ) q^{21} + 6 q^{22} + 3 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{24} + ( -1 + \zeta_{6} ) q^{25} -4 \zeta_{6} q^{26} -5 q^{27} + ( 2 + \zeta_{6} ) q^{28} -3 q^{29} -\zeta_{6} q^{30} + ( -8 + 8 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} + 6 \zeta_{6} q^{33} + ( 3 - 2 \zeta_{6} ) q^{35} -2 q^{36} + 4 \zeta_{6} q^{37} + ( 2 - 2 \zeta_{6} ) q^{38} + ( 4 - 4 \zeta_{6} ) q^{39} -\zeta_{6} q^{40} + 9 q^{41} + ( -1 + 3 \zeta_{6} ) q^{42} -7 q^{43} + 6 \zeta_{6} q^{44} + ( -2 + 2 \zeta_{6} ) q^{45} + ( -3 + 3 \zeta_{6} ) q^{46} + q^{48} + ( -8 + 3 \zeta_{6} ) q^{49} - q^{50} + ( 4 - 4 \zeta_{6} ) q^{52} + ( 6 - 6 \zeta_{6} ) q^{53} -5 \zeta_{6} q^{54} + 6 q^{55} + ( -1 + 3 \zeta_{6} ) q^{56} + 2 q^{57} -3 \zeta_{6} q^{58} + ( 6 - 6 \zeta_{6} ) q^{59} + ( 1 - \zeta_{6} ) q^{60} -5 \zeta_{6} q^{61} -8 q^{62} + ( 6 - 4 \zeta_{6} ) q^{63} + q^{64} -4 \zeta_{6} q^{65} + ( -6 + 6 \zeta_{6} ) q^{66} + ( -5 + 5 \zeta_{6} ) q^{67} -3 q^{69} + ( 2 + \zeta_{6} ) q^{70} -6 q^{71} -2 \zeta_{6} q^{72} + ( 16 - 16 \zeta_{6} ) q^{73} + ( -4 + 4 \zeta_{6} ) q^{74} -\zeta_{6} q^{75} + 2 q^{76} + ( -12 - 6 \zeta_{6} ) q^{77} + 4 q^{78} -2 \zeta_{6} q^{79} + ( 1 - \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + 9 \zeta_{6} q^{82} + 3 q^{83} + ( -3 + 2 \zeta_{6} ) q^{84} -7 \zeta_{6} q^{86} + ( 3 - 3 \zeta_{6} ) q^{87} + ( -6 + 6 \zeta_{6} ) q^{88} + 15 \zeta_{6} q^{89} -2 q^{90} + ( -4 + 12 \zeta_{6} ) q^{91} -3 q^{92} -8 \zeta_{6} q^{93} + ( 2 - 2 \zeta_{6} ) q^{95} + \zeta_{6} q^{96} + 14 q^{97} + ( -3 - 5 \zeta_{6} ) q^{98} + 12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{3} - q^{4} + q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + O(q^{10})$$ $$2 q + q^{2} - q^{3} - q^{4} + q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} - q^{10} + 6 q^{11} - q^{12} - 8 q^{13} + 4 q^{14} - 2 q^{15} - q^{16} - 2 q^{18} - 2 q^{19} - 2 q^{20} + 5 q^{21} + 12 q^{22} + 3 q^{23} + q^{24} - q^{25} - 4 q^{26} - 10 q^{27} + 5 q^{28} - 6 q^{29} - q^{30} - 8 q^{31} + q^{32} + 6 q^{33} + 4 q^{35} - 4 q^{36} + 4 q^{37} + 2 q^{38} + 4 q^{39} - q^{40} + 18 q^{41} + q^{42} - 14 q^{43} + 6 q^{44} - 2 q^{45} - 3 q^{46} + 2 q^{48} - 13 q^{49} - 2 q^{50} + 4 q^{52} + 6 q^{53} - 5 q^{54} + 12 q^{55} + q^{56} + 4 q^{57} - 3 q^{58} + 6 q^{59} + q^{60} - 5 q^{61} - 16 q^{62} + 8 q^{63} + 2 q^{64} - 4 q^{65} - 6 q^{66} - 5 q^{67} - 6 q^{69} + 5 q^{70} - 12 q^{71} - 2 q^{72} + 16 q^{73} - 4 q^{74} - q^{75} + 4 q^{76} - 30 q^{77} + 8 q^{78} - 2 q^{79} + q^{80} - q^{81} + 9 q^{82} + 6 q^{83} - 4 q^{84} - 7 q^{86} + 3 q^{87} - 6 q^{88} + 15 q^{89} - 4 q^{90} + 4 q^{91} - 6 q^{92} - 8 q^{93} + 2 q^{95} + q^{96} + 28 q^{97} - 11 q^{98} + 24 q^{99} + O(q^{100})$$

## 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 −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i −1.00000 −0.500000 2.59808i −1.00000 1.00000 + 1.73205i −0.500000 + 0.866025i
51.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i −1.00000 −0.500000 + 2.59808i −1.00000 1.00000 1.73205i −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.c 2
3.b odd 2 1 630.2.k.b 2
4.b odd 2 1 560.2.q.g 2
5.b even 2 1 350.2.e.e 2
5.c odd 4 2 350.2.j.b 4
7.b odd 2 1 490.2.e.h 2
7.c even 3 1 inner 70.2.e.c 2
7.c even 3 1 490.2.a.c 1
7.d odd 6 1 490.2.a.b 1
7.d odd 6 1 490.2.e.h 2
21.g even 6 1 4410.2.a.bd 1
21.h odd 6 1 630.2.k.b 2
21.h odd 6 1 4410.2.a.bm 1
28.f even 6 1 3920.2.a.bc 1
28.g odd 6 1 560.2.q.g 2
28.g odd 6 1 3920.2.a.p 1
35.i odd 6 1 2450.2.a.bc 1
35.j even 6 1 350.2.e.e 2
35.j even 6 1 2450.2.a.w 1
35.k even 12 2 2450.2.c.l 2
35.l odd 12 2 350.2.j.b 4
35.l odd 12 2 2450.2.c.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.c 2 1.a even 1 1 trivial
70.2.e.c 2 7.c even 3 1 inner
350.2.e.e 2 5.b even 2 1
350.2.e.e 2 35.j even 6 1
350.2.j.b 4 5.c odd 4 2
350.2.j.b 4 35.l odd 12 2
490.2.a.b 1 7.d odd 6 1
490.2.a.c 1 7.c even 3 1
490.2.e.h 2 7.b odd 2 1
490.2.e.h 2 7.d odd 6 1
560.2.q.g 2 4.b odd 2 1
560.2.q.g 2 28.g odd 6 1
630.2.k.b 2 3.b odd 2 1
630.2.k.b 2 21.h odd 6 1
2450.2.a.w 1 35.j even 6 1
2450.2.a.bc 1 35.i odd 6 1
2450.2.c.g 2 35.l odd 12 2
2450.2.c.l 2 35.k even 12 2
3920.2.a.p 1 28.g odd 6 1
3920.2.a.bc 1 28.f even 6 1
4410.2.a.bd 1 21.g even 6 1
4410.2.a.bm 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 $$S_{2}^{\mathrm{new}}(70, [\chi])$$:

 $$T_{3}^{2} + T_{3} + 1$$ $$T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$36 - 6 T + T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$9 - 3 T + T^{2}$$
$29$ $$( 3 + T )^{2}$$
$31$ $$64 + 8 T + T^{2}$$
$37$ $$16 - 4 T + T^{2}$$
$41$ $$( -9 + T )^{2}$$
$43$ $$( 7 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$36 - 6 T + T^{2}$$
$61$ $$25 + 5 T + T^{2}$$
$67$ $$25 + 5 T + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$256 - 16 T + T^{2}$$
$79$ $$4 + 2 T + T^{2}$$
$83$ $$( -3 + T )^{2}$$
$89$ $$225 - 15 T + T^{2}$$
$97$ $$( -14 + T )^{2}$$