# Properties

 Label 210.2.n.a Level $210$ Weight $2$ Character orbit 210.n Analytic conductor $1.677$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 210.n (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.133975 + 2.23205i −1.00000 1.73205 2.00000i 1.00000i 0.500000 0.866025i 1.23205 1.86603i
79.2 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.86603 + 1.23205i −1.00000 −1.73205 + 2.00000i 1.00000i 0.500000 0.866025i −2.23205 + 0.133975i
109.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −0.133975 2.23205i −1.00000 1.73205 + 2.00000i 1.00000i 0.500000 + 0.866025i 1.23205 + 1.86603i
109.2 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i −1.86603 1.23205i −1.00000 −1.73205 2.00000i 1.00000i 0.500000 + 0.866025i −2.23205 0.133975i
 $$n$$: 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.n.a 4
3.b odd 2 1 630.2.u.c 4
4.b odd 2 1 1680.2.di.a 4
5.b even 2 1 inner 210.2.n.a 4
5.c odd 4 1 1050.2.i.f 2
5.c odd 4 1 1050.2.i.o 2
7.b odd 2 1 1470.2.n.i 4
7.c even 3 1 inner 210.2.n.a 4
7.c even 3 1 1470.2.g.f 2
7.d odd 6 1 1470.2.g.a 2
7.d odd 6 1 1470.2.n.i 4
15.d odd 2 1 630.2.u.c 4
20.d odd 2 1 1680.2.di.a 4
21.h odd 6 1 630.2.u.c 4
28.g odd 6 1 1680.2.di.a 4
35.c odd 2 1 1470.2.n.i 4
35.i odd 6 1 1470.2.g.a 2
35.i odd 6 1 1470.2.n.i 4
35.j even 6 1 inner 210.2.n.a 4
35.j even 6 1 1470.2.g.f 2
35.k even 12 1 7350.2.a.b 1
35.k even 12 1 7350.2.a.ch 1
35.l odd 12 1 1050.2.i.f 2
35.l odd 12 1 1050.2.i.o 2
35.l odd 12 1 7350.2.a.t 1
35.l odd 12 1 7350.2.a.bn 1
105.o odd 6 1 630.2.u.c 4
140.p odd 6 1 1680.2.di.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.a 4 1.a even 1 1 trivial
210.2.n.a 4 5.b even 2 1 inner
210.2.n.a 4 7.c even 3 1 inner
210.2.n.a 4 35.j even 6 1 inner
630.2.u.c 4 3.b odd 2 1
630.2.u.c 4 15.d odd 2 1
630.2.u.c 4 21.h odd 6 1
630.2.u.c 4 105.o odd 6 1
1050.2.i.f 2 5.c odd 4 1
1050.2.i.f 2 35.l odd 12 1
1050.2.i.o 2 5.c odd 4 1
1050.2.i.o 2 35.l odd 12 1
1470.2.g.a 2 7.d odd 6 1
1470.2.g.a 2 35.i odd 6 1
1470.2.g.f 2 7.c even 3 1
1470.2.g.f 2 35.j even 6 1
1470.2.n.i 4 7.b odd 2 1
1470.2.n.i 4 7.d odd 6 1
1470.2.n.i 4 35.c odd 2 1
1470.2.n.i 4 35.i odd 6 1
1680.2.di.a 4 4.b odd 2 1
1680.2.di.a 4 20.d odd 2 1
1680.2.di.a 4 28.g odd 6 1
1680.2.di.a 4 140.p odd 6 1
7350.2.a.b 1 35.k even 12 1
7350.2.a.t 1 35.l odd 12 1
7350.2.a.bn 1 35.l odd 12 1
7350.2.a.ch 1 35.k even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} - 5 T_{11} + 25$$ acting on $$S_{2}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$25 + 20 T + 11 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$49 + 2 T^{2} + T^{4}$$
$11$ $$( 25 - 5 T + T^{2} )^{2}$$
$13$ $$( 1 + T^{2} )^{2}$$
$17$ $$16 - 4 T^{2} + T^{4}$$
$19$ $$( 49 - 7 T + T^{2} )^{2}$$
$23$ $$81 - 9 T^{2} + T^{4}$$
$29$ $$T^{4}$$
$31$ $$( 36 - 6 T + T^{2} )^{2}$$
$37$ $$625 - 25 T^{2} + T^{4}$$
$41$ $$( 9 + T )^{4}$$
$43$ $$( 100 + T^{2} )^{2}$$
$47$ $$28561 - 169 T^{2} + T^{4}$$
$53$ $$1 - T^{2} + T^{4}$$
$59$ $$( 16 - 4 T + T^{2} )^{2}$$
$61$ $$( 4 - 2 T + T^{2} )^{2}$$
$67$ $$1296 - 36 T^{2} + T^{4}$$
$71$ $$( 2 + T )^{4}$$
$73$ $$256 - 16 T^{2} + T^{4}$$
$79$ $$( 196 + 14 T + T^{2} )^{2}$$
$83$ $$( 100 + T^{2} )^{2}$$
$89$ $$( 100 - 10 T + T^{2} )^{2}$$
$97$ $$( 64 + T^{2} )^{2}$$