# Properties

 Label 630.2.j.e Level 630 Weight 2 Character orbit 630.j Analytic conductor 5.031 Analytic rank 0 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: $$0$$ 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.

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

## 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 1.50000 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i 1.50000 + 0.866025i −0.500000 0.866025i −1.00000 1.50000 2.59808i 1.00000
421.1 0.500000 0.866025i 1.50000 + 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i 1.50000 0.866025i −0.500000 + 0.866025i −1.00000 1.50000 + 2.59808i 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 $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{2} + 5 T_{11} + 25$$ $$T_{13}^{2} - 2 T_{13} + 4$$