# Properties

 Label 210.2.i.a Level $210$ Weight $2$ Character orbit 210.i Analytic conductor $1.677$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 210.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + q^{6} + ( 3 - 2 \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + q^{6} + ( 3 - 2 \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{10} + ( 1 - \zeta_{6} ) q^{11} -\zeta_{6} q^{12} + 7 q^{13} + ( -2 - \zeta_{6} ) q^{14} + q^{15} -\zeta_{6} q^{16} + ( 4 - 4 \zeta_{6} ) q^{17} + ( -1 + \zeta_{6} ) q^{18} -\zeta_{6} q^{19} + q^{20} + ( -1 + 3 \zeta_{6} ) q^{21} - q^{22} -\zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{24} + ( -1 + \zeta_{6} ) q^{25} -7 \zeta_{6} q^{26} + q^{27} + ( -1 + 3 \zeta_{6} ) q^{28} -8 q^{29} -\zeta_{6} q^{30} + ( -6 + 6 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + \zeta_{6} q^{33} -4 q^{34} + ( -2 - \zeta_{6} ) q^{35} + q^{36} + 3 \zeta_{6} q^{37} + ( -1 + \zeta_{6} ) q^{38} + ( -7 + 7 \zeta_{6} ) q^{39} -\zeta_{6} q^{40} + 9 q^{41} + ( 3 - 2 \zeta_{6} ) q^{42} -4 q^{43} + \zeta_{6} q^{44} + ( -1 + \zeta_{6} ) q^{45} + ( -1 + \zeta_{6} ) q^{46} + 3 \zeta_{6} q^{47} + q^{48} + ( 5 - 8 \zeta_{6} ) q^{49} + q^{50} + 4 \zeta_{6} q^{51} + ( -7 + 7 \zeta_{6} ) q^{52} + ( 1 - \zeta_{6} ) q^{53} -\zeta_{6} q^{54} - q^{55} + ( 3 - 2 \zeta_{6} ) q^{56} + q^{57} + 8 \zeta_{6} q^{58} + ( -12 + 12 \zeta_{6} ) q^{59} + ( -1 + \zeta_{6} ) q^{60} + 4 \zeta_{6} q^{61} + 6 q^{62} + ( -2 - \zeta_{6} ) q^{63} + q^{64} -7 \zeta_{6} q^{65} + ( 1 - \zeta_{6} ) q^{66} + ( -12 + 12 \zeta_{6} ) q^{67} + 4 \zeta_{6} q^{68} + q^{69} + ( -1 + 3 \zeta_{6} ) q^{70} -14 q^{71} -\zeta_{6} q^{72} + ( 14 - 14 \zeta_{6} ) q^{73} + ( 3 - 3 \zeta_{6} ) q^{74} -\zeta_{6} q^{75} + q^{76} + ( 1 - 3 \zeta_{6} ) q^{77} + 7 q^{78} -4 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -9 \zeta_{6} q^{82} + 12 q^{83} + ( -2 - \zeta_{6} ) q^{84} -4 q^{85} + 4 \zeta_{6} q^{86} + ( 8 - 8 \zeta_{6} ) q^{87} + ( 1 - \zeta_{6} ) q^{88} + 2 \zeta_{6} q^{89} + q^{90} + ( 21 - 14 \zeta_{6} ) q^{91} + q^{92} -6 \zeta_{6} q^{93} + ( 3 - 3 \zeta_{6} ) q^{94} + ( -1 + \zeta_{6} ) q^{95} -\zeta_{6} q^{96} -16 q^{97} + ( -8 + 3 \zeta_{6} ) q^{98} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{3} - q^{4} - q^{5} + 2q^{6} + 4q^{7} + 2q^{8} - q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{3} - q^{4} - q^{5} + 2q^{6} + 4q^{7} + 2q^{8} - q^{9} - q^{10} + q^{11} - q^{12} + 14q^{13} - 5q^{14} + 2q^{15} - q^{16} + 4q^{17} - q^{18} - q^{19} + 2q^{20} + q^{21} - 2q^{22} - q^{23} - q^{24} - q^{25} - 7q^{26} + 2q^{27} + q^{28} - 16q^{29} - q^{30} - 6q^{31} - q^{32} + q^{33} - 8q^{34} - 5q^{35} + 2q^{36} + 3q^{37} - q^{38} - 7q^{39} - q^{40} + 18q^{41} + 4q^{42} - 8q^{43} + q^{44} - q^{45} - q^{46} + 3q^{47} + 2q^{48} + 2q^{49} + 2q^{50} + 4q^{51} - 7q^{52} + q^{53} - q^{54} - 2q^{55} + 4q^{56} + 2q^{57} + 8q^{58} - 12q^{59} - q^{60} + 4q^{61} + 12q^{62} - 5q^{63} + 2q^{64} - 7q^{65} + q^{66} - 12q^{67} + 4q^{68} + 2q^{69} + q^{70} - 28q^{71} - q^{72} + 14q^{73} + 3q^{74} - q^{75} + 2q^{76} - q^{77} + 14q^{78} - 4q^{79} - q^{80} - q^{81} - 9q^{82} + 24q^{83} - 5q^{84} - 8q^{85} + 4q^{86} + 8q^{87} + q^{88} + 2q^{89} + 2q^{90} + 28q^{91} + 2q^{92} - 6q^{93} + 3q^{94} - q^{95} - q^{96} - 32q^{97} - 13q^{98} - 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/210\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 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 2.00000 + 1.73205i 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
151.1 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000 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 210.2.i.a 2
3.b odd 2 1 630.2.k.h 2
4.b odd 2 1 1680.2.bg.k 2
5.b even 2 1 1050.2.i.s 2
5.c odd 4 2 1050.2.o.j 4
7.b odd 2 1 1470.2.i.i 2
7.c even 3 1 inner 210.2.i.a 2
7.c even 3 1 1470.2.a.r 1
7.d odd 6 1 1470.2.a.k 1
7.d odd 6 1 1470.2.i.i 2
21.g even 6 1 4410.2.a.q 1
21.h odd 6 1 630.2.k.h 2
21.h odd 6 1 4410.2.a.g 1
28.g odd 6 1 1680.2.bg.k 2
35.i odd 6 1 7350.2.a.ba 1
35.j even 6 1 1050.2.i.s 2
35.j even 6 1 7350.2.a.j 1
35.l odd 12 2 1050.2.o.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.a 2 1.a even 1 1 trivial
210.2.i.a 2 7.c even 3 1 inner
630.2.k.h 2 3.b odd 2 1
630.2.k.h 2 21.h odd 6 1
1050.2.i.s 2 5.b even 2 1
1050.2.i.s 2 35.j even 6 1
1050.2.o.j 4 5.c odd 4 2
1050.2.o.j 4 35.l odd 12 2
1470.2.a.k 1 7.d odd 6 1
1470.2.a.r 1 7.c even 3 1
1470.2.i.i 2 7.b odd 2 1
1470.2.i.i 2 7.d odd 6 1
1680.2.bg.k 2 4.b odd 2 1
1680.2.bg.k 2 28.g odd 6 1
4410.2.a.g 1 21.h odd 6 1
4410.2.a.q 1 21.g even 6 1
7350.2.a.j 1 35.j even 6 1
7350.2.a.ba 1 35.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(210, [\chi])$$:

 $$T_{11}^{2} - T_{11} + 1$$ $$T_{13} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$7 - 4 T + T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$( -7 + T )^{2}$$
$17$ $$16 - 4 T + T^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$1 + T + T^{2}$$
$29$ $$( 8 + T )^{2}$$
$31$ $$36 + 6 T + T^{2}$$
$37$ $$9 - 3 T + T^{2}$$
$41$ $$( -9 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$9 - 3 T + T^{2}$$
$53$ $$1 - T + T^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$16 - 4 T + T^{2}$$
$67$ $$144 + 12 T + T^{2}$$
$71$ $$( 14 + T )^{2}$$
$73$ $$196 - 14 T + T^{2}$$
$79$ $$16 + 4 T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$4 - 2 T + T^{2}$$
$97$ $$( 16 + T )^{2}$$