# Properties

 Label 210.2.g.a Level $210$ Weight $2$ Character orbit 210.g 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.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 −1.00000 2.00000i −1.00000 1.00000i 1.00000i −1.00000 −2.00000 + 1.00000i
169.2 1.00000i 1.00000i −1.00000 −1.00000 + 2.00000i −1.00000 1.00000i 1.00000i −1.00000 −2.00000 1.00000i
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.g.a 2
3.b odd 2 1 630.2.g.d 2
4.b odd 2 1 1680.2.t.d 2
5.b even 2 1 inner 210.2.g.a 2
5.c odd 4 1 1050.2.a.g 1
5.c odd 4 1 1050.2.a.m 1
7.b odd 2 1 1470.2.g.e 2
7.c even 3 2 1470.2.n.g 4
7.d odd 6 2 1470.2.n.c 4
12.b even 2 1 5040.2.t.k 2
15.d odd 2 1 630.2.g.d 2
15.e even 4 1 3150.2.a.q 1
15.e even 4 1 3150.2.a.be 1
20.d odd 2 1 1680.2.t.d 2
20.e even 4 1 8400.2.a.bd 1
20.e even 4 1 8400.2.a.ca 1
35.c odd 2 1 1470.2.g.e 2
35.f even 4 1 7350.2.a.g 1
35.f even 4 1 7350.2.a.co 1
35.i odd 6 2 1470.2.n.c 4
35.j even 6 2 1470.2.n.g 4
60.h even 2 1 5040.2.t.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.g.a 2 1.a even 1 1 trivial
210.2.g.a 2 5.b even 2 1 inner
630.2.g.d 2 3.b odd 2 1
630.2.g.d 2 15.d odd 2 1
1050.2.a.g 1 5.c odd 4 1
1050.2.a.m 1 5.c odd 4 1
1470.2.g.e 2 7.b odd 2 1
1470.2.g.e 2 35.c odd 2 1
1470.2.n.c 4 7.d odd 6 2
1470.2.n.c 4 35.i odd 6 2
1470.2.n.g 4 7.c even 3 2
1470.2.n.g 4 35.j even 6 2
1680.2.t.d 2 4.b odd 2 1
1680.2.t.d 2 20.d odd 2 1
3150.2.a.q 1 15.e even 4 1
3150.2.a.be 1 15.e even 4 1
5040.2.t.k 2 12.b even 2 1
5040.2.t.k 2 60.h even 2 1
7350.2.a.g 1 35.f even 4 1
7350.2.a.co 1 35.f even 4 1
8400.2.a.bd 1 20.e even 4 1
8400.2.a.ca 1 20.e even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$5 + 2 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$64 + T^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$( -6 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$16 + T^{2}$$
$53$ $$4 + T^{2}$$
$59$ $$( -8 + T )^{2}$$
$61$ $$( -10 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$( 14 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$256 + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$100 + T^{2}$$