# Properties

 Label 210.2.i.c 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.

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

## 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.c 2
3.b odd 2 1 630.2.k.a 2
4.b odd 2 1 1680.2.bg.n 2
5.b even 2 1 1050.2.i.i 2
5.c odd 4 2 1050.2.o.c 4
7.b odd 2 1 1470.2.i.p 2
7.c even 3 1 inner 210.2.i.c 2
7.c even 3 1 1470.2.a.f 1
7.d odd 6 1 1470.2.a.e 1
7.d odd 6 1 1470.2.i.p 2
21.g even 6 1 4410.2.a.w 1
21.h odd 6 1 630.2.k.a 2
21.h odd 6 1 4410.2.a.bh 1
28.g odd 6 1 1680.2.bg.n 2
35.i odd 6 1 7350.2.a.cx 1
35.j even 6 1 1050.2.i.i 2
35.j even 6 1 7350.2.a.cd 1
35.l odd 12 2 1050.2.o.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.c 2 1.a even 1 1 trivial
210.2.i.c 2 7.c even 3 1 inner
630.2.k.a 2 3.b odd 2 1
630.2.k.a 2 21.h odd 6 1
1050.2.i.i 2 5.b even 2 1
1050.2.i.i 2 35.j even 6 1
1050.2.o.c 4 5.c odd 4 2
1050.2.o.c 4 35.l odd 12 2
1470.2.a.e 1 7.d odd 6 1
1470.2.a.f 1 7.c even 3 1
1470.2.i.p 2 7.b odd 2 1
1470.2.i.p 2 7.d odd 6 1
1680.2.bg.n 2 4.b odd 2 1
1680.2.bg.n 2 28.g odd 6 1
4410.2.a.w 1 21.g even 6 1
4410.2.a.bh 1 21.h odd 6 1
7350.2.a.cd 1 35.j even 6 1
7350.2.a.cx 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 $$S_{2}^{\mathrm{new}}(210, [\chi])$$:

 $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{13} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$( -5 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$25 + 5 T + T^{2}$$
$23$ $$81 - 9 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$100 - 10 T + T^{2}$$
$37$ $$1 - T + T^{2}$$
$41$ $$( -9 + T )^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$9 + 3 T + T^{2}$$
$53$ $$9 - 3 T + T^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$64 + 8 T + T^{2}$$
$67$ $$64 + 8 T + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$4 + 2 T + T^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( -8 + T )^{2}$$