# Properties

 Label 210.2.i.b 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.

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

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

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

## Hecke kernels

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

 $$T_{11}^{2} - 5 T_{11} + 25$$ $$T_{13} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$25 - 5 T + T^{2}$$
$13$ $$( 5 + T )^{2}$$
$17$ $$16 - 4 T + T^{2}$$
$19$ $$49 - 7 T + T^{2}$$
$23$ $$1 + T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$4 - 2 T + T^{2}$$
$37$ $$1 + T + T^{2}$$
$41$ $$( -5 + T )^{2}$$
$43$ $$( -12 + T )^{2}$$
$47$ $$121 - 11 T + T^{2}$$
$53$ $$81 - 9 T + T^{2}$$
$59$ $$16 + 4 T + T^{2}$$
$61$ $$16 + 4 T + T^{2}$$
$67$ $$144 - 12 T + T^{2}$$
$71$ $$( -2 + T )^{2}$$
$73$ $$100 + 10 T + T^{2}$$
$79$ $$144 - 12 T + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$196 + 14 T + T^{2}$$
$97$ $$( 8 + T )^{2}$$