# Properties

 Label 630.2.j.a Level 630 Weight 2 Character orbit 630.j Analytic conductor 5.031 Analytic rank 1 Dimension 2 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 630.j (of order $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$5.03057532734$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( -1 + 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{5} + ( -1 - \zeta_{6} ) q^{6} + ( -1 + \zeta_{6} ) q^{7} + q^{8} -3 q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( -1 + 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{5} + ( -1 - \zeta_{6} ) q^{6} + ( -1 + \zeta_{6} ) q^{7} + q^{8} -3 q^{9} + q^{10} + ( -3 + 3 \zeta_{6} ) q^{11} + ( 2 - \zeta_{6} ) q^{12} -5 \zeta_{6} q^{13} -\zeta_{6} q^{14} + ( 2 - \zeta_{6} ) q^{15} + ( -1 + \zeta_{6} ) q^{16} -3 q^{17} + ( 3 - 3 \zeta_{6} ) q^{18} + 2 q^{19} + ( -1 + \zeta_{6} ) q^{20} + ( -1 - \zeta_{6} ) q^{21} -3 \zeta_{6} q^{22} -6 \zeta_{6} q^{23} + ( -1 + 2 \zeta_{6} ) q^{24} + ( -1 + \zeta_{6} ) q^{25} + 5 q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} + ( -1 + 2 \zeta_{6} ) q^{30} -2 \zeta_{6} q^{31} -\zeta_{6} q^{32} + ( -3 - 3 \zeta_{6} ) q^{33} + ( 3 - 3 \zeta_{6} ) q^{34} + q^{35} + 3 \zeta_{6} q^{36} -10 q^{37} + ( -2 + 2 \zeta_{6} ) q^{38} + ( 10 - 5 \zeta_{6} ) q^{39} -\zeta_{6} q^{40} + ( 2 - \zeta_{6} ) q^{42} + ( 10 - 10 \zeta_{6} ) q^{43} + 3 q^{44} + 3 \zeta_{6} q^{45} + 6 q^{46} + ( -3 + 3 \zeta_{6} ) q^{47} + ( -1 - \zeta_{6} ) q^{48} -\zeta_{6} q^{49} -\zeta_{6} q^{50} + ( 3 - 6 \zeta_{6} ) q^{51} + ( -5 + 5 \zeta_{6} ) q^{52} + ( 3 + 3 \zeta_{6} ) q^{54} + 3 q^{55} + ( -1 + \zeta_{6} ) q^{56} + ( -2 + 4 \zeta_{6} ) q^{57} -6 \zeta_{6} q^{58} + 12 \zeta_{6} q^{59} + ( -1 - \zeta_{6} ) q^{60} + ( 10 - 10 \zeta_{6} ) q^{61} + 2 q^{62} + ( 3 - 3 \zeta_{6} ) q^{63} + q^{64} + ( -5 + 5 \zeta_{6} ) q^{65} + ( 6 - 3 \zeta_{6} ) q^{66} -2 \zeta_{6} q^{67} + 3 \zeta_{6} q^{68} + ( 12 - 6 \zeta_{6} ) q^{69} + ( -1 + \zeta_{6} ) q^{70} -9 q^{71} -3 q^{72} + 11 q^{73} + ( 10 - 10 \zeta_{6} ) q^{74} + ( -1 - \zeta_{6} ) q^{75} -2 \zeta_{6} q^{76} -3 \zeta_{6} q^{77} + ( -5 + 10 \zeta_{6} ) q^{78} + ( 1 - \zeta_{6} ) q^{79} + q^{80} + 9 q^{81} + ( -9 + 9 \zeta_{6} ) q^{83} + ( -1 + 2 \zeta_{6} ) q^{84} + 3 \zeta_{6} q^{85} + 10 \zeta_{6} q^{86} + ( -6 - 6 \zeta_{6} ) q^{87} + ( -3 + 3 \zeta_{6} ) q^{88} -6 q^{89} -3 q^{90} + 5 q^{91} + ( -6 + 6 \zeta_{6} ) q^{92} + ( 4 - 2 \zeta_{6} ) q^{93} -3 \zeta_{6} q^{94} -2 \zeta_{6} q^{95} + ( 2 - \zeta_{6} ) q^{96} + ( -17 + 17 \zeta_{6} ) q^{97} + q^{98} + ( 9 - 9 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} - q^{5} - 3q^{6} - q^{7} + 2q^{8} - 6q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{4} - q^{5} - 3q^{6} - q^{7} + 2q^{8} - 6q^{9} + 2q^{10} - 3q^{11} + 3q^{12} - 5q^{13} - q^{14} + 3q^{15} - q^{16} - 6q^{17} + 3q^{18} + 4q^{19} - q^{20} - 3q^{21} - 3q^{22} - 6q^{23} - q^{25} + 10q^{26} + 2q^{28} - 6q^{29} - 2q^{31} - q^{32} - 9q^{33} + 3q^{34} + 2q^{35} + 3q^{36} - 20q^{37} - 2q^{38} + 15q^{39} - q^{40} + 3q^{42} + 10q^{43} + 6q^{44} + 3q^{45} + 12q^{46} - 3q^{47} - 3q^{48} - q^{49} - q^{50} - 5q^{52} + 9q^{54} + 6q^{55} - q^{56} - 6q^{58} + 12q^{59} - 3q^{60} + 10q^{61} + 4q^{62} + 3q^{63} + 2q^{64} - 5q^{65} + 9q^{66} - 2q^{67} + 3q^{68} + 18q^{69} - q^{70} - 18q^{71} - 6q^{72} + 22q^{73} + 10q^{74} - 3q^{75} - 2q^{76} - 3q^{77} + q^{79} + 2q^{80} + 18q^{81} - 9q^{83} + 3q^{85} + 10q^{86} - 18q^{87} - 3q^{88} - 12q^{89} - 6q^{90} + 10q^{91} - 6q^{92} + 6q^{93} - 3q^{94} - 2q^{95} + 3q^{96} - 17q^{97} + 2q^{98} + 9q^{99} + O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/630\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
211.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 1.73205i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.50000 + 0.866025i −0.500000 0.866025i 1.00000 −3.00000 1.00000
421.1 −0.500000 + 0.866025i 1.73205i −0.500000 0.866025i −0.500000 0.866025i −1.50000 0.866025i −0.500000 + 0.866025i 1.00000 −3.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
9.c Even 1 yes

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{13}^{2} + 5 T_{13} + 25$$