# Properties

 Label 210.2.i.d 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.

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

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

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4}$$
$13$ $$( 1 - T + 13 T^{2} )^{2}$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$1 - 3 T - 10 T^{2} - 57 T^{3} + 361 T^{4}$$
$23$ $$1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 8 T + 29 T^{2} )^{2}$$
$31$ $$1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4}$$
$37$ $$( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$
$41$ $$( 1 + 11 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{2}$$
$47$ $$1 - 5 T - 22 T^{2} - 235 T^{3} + 2209 T^{4}$$
$53$ $$1 - 11 T + 68 T^{2} - 583 T^{3} + 2809 T^{4}$$
$59$ $$1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4}$$
$61$ $$1 - 61 T^{2} + 3721 T^{4}$$
$67$ $$1 - 67 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 6 T + 71 T^{2} )^{2}$$
$73$ $$1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4}$$
$79$ $$1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 8 T + 83 T^{2} )^{2}$$
$89$ $$1 - 10 T + 11 T^{2} - 890 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 16 T + 97 T^{2} )^{2}$$