# Properties

 Label 630.2.i.d 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( 2 - \zeta_{6} ) q^{3} + q^{4} + ( 1 - \zeta_{6} ) q^{5} + ( 2 - \zeta_{6} ) q^{6} + ( -1 - 2 \zeta_{6} ) q^{7} + q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + q^{2} + ( 2 - \zeta_{6} ) q^{3} + q^{4} + ( 1 - \zeta_{6} ) q^{5} + ( 2 - \zeta_{6} ) q^{6} + ( -1 - 2 \zeta_{6} ) q^{7} + q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} + ( 1 - \zeta_{6} ) q^{10} + ( 2 - \zeta_{6} ) q^{12} + 4 \zeta_{6} q^{13} + ( -1 - 2 \zeta_{6} ) q^{14} + ( 1 - 2 \zeta_{6} ) q^{15} + q^{16} + ( 3 - 3 \zeta_{6} ) q^{18} -2 \zeta_{6} q^{19} + ( 1 - \zeta_{6} ) q^{20} + ( -4 - \zeta_{6} ) q^{21} + ( 3 - 3 \zeta_{6} ) q^{23} + ( 2 - \zeta_{6} ) q^{24} -\zeta_{6} q^{25} + 4 \zeta_{6} q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -1 - 2 \zeta_{6} ) q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} + ( 1 - 2 \zeta_{6} ) q^{30} + 2 q^{31} + q^{32} + ( -3 + \zeta_{6} ) q^{35} + ( 3 - 3 \zeta_{6} ) q^{36} -2 \zeta_{6} q^{37} -2 \zeta_{6} q^{38} + ( 4 + 4 \zeta_{6} ) q^{39} + ( 1 - \zeta_{6} ) q^{40} + 6 \zeta_{6} q^{41} + ( -4 - \zeta_{6} ) q^{42} + ( -11 + 11 \zeta_{6} ) q^{43} -3 \zeta_{6} q^{45} + ( 3 - 3 \zeta_{6} ) q^{46} + 3 q^{47} + ( 2 - \zeta_{6} ) q^{48} + ( -3 + 8 \zeta_{6} ) q^{49} -\zeta_{6} q^{50} + 4 \zeta_{6} q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} + ( 3 - 6 \zeta_{6} ) q^{54} + ( -1 - 2 \zeta_{6} ) q^{56} + ( -2 - 2 \zeta_{6} ) q^{57} + ( -6 + 6 \zeta_{6} ) q^{58} + 12 q^{59} + ( 1 - 2 \zeta_{6} ) q^{60} -7 q^{61} + 2 q^{62} + ( -9 + 3 \zeta_{6} ) q^{63} + q^{64} + 4 q^{65} -7 q^{67} + ( 3 - 6 \zeta_{6} ) q^{69} + ( -3 + \zeta_{6} ) q^{70} + ( 3 - 3 \zeta_{6} ) q^{72} + ( -2 + 2 \zeta_{6} ) q^{73} -2 \zeta_{6} q^{74} + ( -1 - \zeta_{6} ) q^{75} -2 \zeta_{6} q^{76} + ( 4 + 4 \zeta_{6} ) q^{78} -4 q^{79} + ( 1 - \zeta_{6} ) q^{80} -9 \zeta_{6} q^{81} + 6 \zeta_{6} q^{82} + ( 12 - 12 \zeta_{6} ) q^{83} + ( -4 - \zeta_{6} ) q^{84} + ( -11 + 11 \zeta_{6} ) q^{86} + ( -6 + 12 \zeta_{6} ) q^{87} + 15 \zeta_{6} q^{89} -3 \zeta_{6} q^{90} + ( 8 - 12 \zeta_{6} ) q^{91} + ( 3 - 3 \zeta_{6} ) q^{92} + ( 4 - 2 \zeta_{6} ) q^{93} + 3 q^{94} -2 q^{95} + ( 2 - \zeta_{6} ) q^{96} + ( 10 - 10 \zeta_{6} ) q^{97} + ( -3 + 8 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 3q^{3} + 2q^{4} + q^{5} + 3q^{6} - 4q^{7} + 2q^{8} + 3q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 3q^{3} + 2q^{4} + q^{5} + 3q^{6} - 4q^{7} + 2q^{8} + 3q^{9} + q^{10} + 3q^{12} + 4q^{13} - 4q^{14} + 2q^{16} + 3q^{18} - 2q^{19} + q^{20} - 9q^{21} + 3q^{23} + 3q^{24} - q^{25} + 4q^{26} - 4q^{28} - 6q^{29} + 4q^{31} + 2q^{32} - 5q^{35} + 3q^{36} - 2q^{37} - 2q^{38} + 12q^{39} + q^{40} + 6q^{41} - 9q^{42} - 11q^{43} - 3q^{45} + 3q^{46} + 6q^{47} + 3q^{48} + 2q^{49} - q^{50} + 4q^{52} - 6q^{53} - 4q^{56} - 6q^{57} - 6q^{58} + 24q^{59} - 14q^{61} + 4q^{62} - 15q^{63} + 2q^{64} + 8q^{65} - 14q^{67} - 5q^{70} + 3q^{72} - 2q^{73} - 2q^{74} - 3q^{75} - 2q^{76} + 12q^{78} - 8q^{79} + q^{80} - 9q^{81} + 6q^{82} + 12q^{83} - 9q^{84} - 11q^{86} + 15q^{89} - 3q^{90} + 4q^{91} + 3q^{92} + 6q^{93} + 6q^{94} - 4q^{95} + 3q^{96} + 10q^{97} + 2q^{98} + O(q^{100})$$

## 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 −2.00000 + 1.73205i 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 −2.00000 1.73205i 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 $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}$$ $$T_{13}^{2} - 4 T_{13} + 16$$