# Properties

 Label 630.2.i.c Level 630 Weight 2 Character orbit 630.i 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.i (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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( 1 + \zeta_{6} ) q^{3} + q^{4} + ( -1 + \zeta_{6} ) q^{5} + ( 1 + \zeta_{6} ) q^{6} + ( -3 + \zeta_{6} ) q^{7} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + q^{2} + ( 1 + \zeta_{6} ) q^{3} + q^{4} + ( -1 + \zeta_{6} ) q^{5} + ( 1 + \zeta_{6} ) q^{6} + ( -3 + \zeta_{6} ) q^{7} + q^{8} + 3 \zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{10} + 2 \zeta_{6} q^{11} + ( 1 + \zeta_{6} ) q^{12} + 2 \zeta_{6} q^{13} + ( -3 + \zeta_{6} ) q^{14} + ( -2 + \zeta_{6} ) q^{15} + q^{16} + 3 \zeta_{6} q^{18} + ( -1 + \zeta_{6} ) q^{20} + ( -4 - \zeta_{6} ) q^{21} + 2 \zeta_{6} q^{22} + ( 4 - 4 \zeta_{6} ) q^{23} + ( 1 + \zeta_{6} ) q^{24} -\zeta_{6} q^{25} + 2 \zeta_{6} q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -3 + \zeta_{6} ) q^{28} + ( 1 - \zeta_{6} ) q^{29} + ( -2 + \zeta_{6} ) q^{30} + 10 q^{31} + q^{32} + ( -2 + 4 \zeta_{6} ) q^{33} + ( 2 - 3 \zeta_{6} ) q^{35} + 3 \zeta_{6} q^{36} -8 \zeta_{6} q^{37} + ( -2 + 4 \zeta_{6} ) q^{39} + ( -1 + \zeta_{6} ) q^{40} -5 \zeta_{6} q^{41} + ( -4 - \zeta_{6} ) q^{42} + ( 1 - \zeta_{6} ) q^{43} + 2 \zeta_{6} q^{44} -3 q^{45} + ( 4 - 4 \zeta_{6} ) q^{46} -13 q^{47} + ( 1 + \zeta_{6} ) q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} -\zeta_{6} q^{50} + 2 \zeta_{6} q^{52} + ( 10 - 10 \zeta_{6} ) q^{53} + ( -3 + 6 \zeta_{6} ) q^{54} -2 q^{55} + ( -3 + \zeta_{6} ) q^{56} + ( 1 - \zeta_{6} ) q^{58} -4 q^{59} + ( -2 + \zeta_{6} ) q^{60} -6 q^{61} + 10 q^{62} + ( -3 - 6 \zeta_{6} ) q^{63} + q^{64} -2 q^{65} + ( -2 + 4 \zeta_{6} ) q^{66} + 12 q^{67} + ( 8 - 4 \zeta_{6} ) q^{69} + ( 2 - 3 \zeta_{6} ) q^{70} -12 q^{71} + 3 \zeta_{6} q^{72} -8 \zeta_{6} q^{74} + ( 1 - 2 \zeta_{6} ) q^{75} + ( -2 - 4 \zeta_{6} ) q^{77} + ( -2 + 4 \zeta_{6} ) q^{78} + 10 q^{79} + ( -1 + \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} -5 \zeta_{6} q^{82} + ( 5 - 5 \zeta_{6} ) q^{83} + ( -4 - \zeta_{6} ) q^{84} + ( 1 - \zeta_{6} ) q^{86} + ( 2 - \zeta_{6} ) q^{87} + 2 \zeta_{6} q^{88} + 14 \zeta_{6} q^{89} -3 q^{90} + ( -2 - 4 \zeta_{6} ) q^{91} + ( 4 - 4 \zeta_{6} ) q^{92} + ( 10 + 10 \zeta_{6} ) q^{93} -13 q^{94} + ( 1 + \zeta_{6} ) q^{96} + ( -2 + 2 \zeta_{6} ) q^{97} + ( 8 - 5 \zeta_{6} ) q^{98} + ( -6 + 6 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 3q^{3} + 2q^{4} - q^{5} + 3q^{6} - 5q^{7} + 2q^{8} + 3q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 3q^{3} + 2q^{4} - q^{5} + 3q^{6} - 5q^{7} + 2q^{8} + 3q^{9} - q^{10} + 2q^{11} + 3q^{12} + 2q^{13} - 5q^{14} - 3q^{15} + 2q^{16} + 3q^{18} - q^{20} - 9q^{21} + 2q^{22} + 4q^{23} + 3q^{24} - q^{25} + 2q^{26} - 5q^{28} + q^{29} - 3q^{30} + 20q^{31} + 2q^{32} + q^{35} + 3q^{36} - 8q^{37} - q^{40} - 5q^{41} - 9q^{42} + q^{43} + 2q^{44} - 6q^{45} + 4q^{46} - 26q^{47} + 3q^{48} + 11q^{49} - q^{50} + 2q^{52} + 10q^{53} - 4q^{55} - 5q^{56} + q^{58} - 8q^{59} - 3q^{60} - 12q^{61} + 20q^{62} - 12q^{63} + 2q^{64} - 4q^{65} + 24q^{67} + 12q^{69} + q^{70} - 24q^{71} + 3q^{72} - 8q^{74} - 8q^{77} + 20q^{79} - q^{80} - 9q^{81} - 5q^{82} + 5q^{83} - 9q^{84} + q^{86} + 3q^{87} + 2q^{88} + 14q^{89} - 6q^{90} - 8q^{91} + 4q^{92} + 30q^{93} - 26q^{94} + 3q^{96} - 2q^{97} + 11q^{98} - 6q^{99} + O(q^{100})$$

## Character Values

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

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

## 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
1.00000 1.50000 0.866025i 1.00000 −0.500000 0.866025i 1.50000 0.866025i −2.50000 0.866025i 1.00000 1.50000 2.59808i −0.500000 0.866025i
151.1 1.00000 1.50000 + 0.866025i 1.00000 −0.500000 + 0.866025i 1.50000 + 0.866025i −2.50000 + 0.866025i 1.00000 1.50000 + 2.59808i −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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
63.h Even 1 yes

## Hecke kernels

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

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