# Properties

 Label 630.2.j.g Level 630 Weight 2 Character orbit 630.j Analytic conductor 5.031 Analytic rank 0 Dimension 4 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} ) q^{2} -\beta_{3} q^{3} -\beta_{2} q^{4} + \beta_{2} q^{5} + ( \beta_{1} - \beta_{3} ) q^{6} + ( 1 - \beta_{2} ) q^{7} - q^{8} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{2} ) q^{2} -\beta_{3} q^{3} -\beta_{2} q^{4} + \beta_{2} q^{5} + ( \beta_{1} - \beta_{3} ) q^{6} + ( 1 - \beta_{2} ) q^{7} - q^{8} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{9} + q^{10} + ( 1 - \beta_{2} ) q^{11} + \beta_{1} q^{12} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{13} -\beta_{2} q^{14} -\beta_{1} q^{15} + ( -1 + \beta_{2} ) q^{16} + 3 q^{17} + ( -3 \beta_{2} - \beta_{3} ) q^{18} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( 1 - \beta_{2} ) q^{20} + ( \beta_{1} - \beta_{3} ) q^{21} -\beta_{2} q^{22} + 2 \beta_{2} q^{23} + \beta_{3} q^{24} + ( -1 + \beta_{2} ) q^{25} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{26} + ( 3 + 2 \beta_{1} - 2 \beta_{3} ) q^{27} - q^{28} + ( 2 - 2 \beta_{2} ) q^{29} -\beta_{3} q^{30} + 2 \beta_{2} q^{31} + \beta_{2} q^{32} + ( \beta_{1} - \beta_{3} ) q^{33} + ( 3 - 3 \beta_{2} ) q^{34} + q^{35} + ( -3 + \beta_{1} - \beta_{3} ) q^{36} + ( 4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{37} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{38} + ( -3 - 3 \beta_{2} - \beta_{3} ) q^{39} -\beta_{2} q^{40} + ( -1 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{41} + \beta_{1} q^{42} + ( -6 - \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{43} - q^{44} + ( 3 - \beta_{1} + \beta_{3} ) q^{45} + 2 q^{46} + ( 3 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{47} + ( -\beta_{1} + \beta_{3} ) q^{48} -\beta_{2} q^{49} + \beta_{2} q^{50} -3 \beta_{3} q^{51} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{52} + ( -6 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 3 + 2 \beta_{1} - 3 \beta_{2} ) q^{54} + q^{55} + ( -1 + \beta_{2} ) q^{56} + ( -6 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{57} -2 \beta_{2} q^{58} + ( 3 - 6 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{59} + ( \beta_{1} - \beta_{3} ) q^{60} + ( 2 - 4 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} ) q^{61} + 2 q^{62} + ( -3 \beta_{2} - \beta_{3} ) q^{63} + q^{64} + ( -1 + \beta_{1} - 2 \beta_{3} ) q^{65} + \beta_{1} q^{66} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{67} -3 \beta_{2} q^{68} -2 \beta_{1} q^{69} + ( 1 - \beta_{2} ) q^{70} + ( -6 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{71} + ( -3 + \beta_{1} + 3 \beta_{2} ) q^{72} + ( 1 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{73} + ( 6 - 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{74} + ( -\beta_{1} + \beta_{3} ) q^{75} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{76} -\beta_{2} q^{77} + ( -6 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{78} + ( -7 - 3 \beta_{1} + 10 \beta_{2} + 6 \beta_{3} ) q^{79} - q^{80} + ( -6 \beta_{2} - 5 \beta_{3} ) q^{81} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{82} + ( -3 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{83} + \beta_{3} q^{84} + 3 \beta_{2} q^{85} + ( 1 - 2 \beta_{1} + 6 \beta_{2} + \beta_{3} ) q^{86} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{87} + ( -1 + \beta_{2} ) q^{88} + ( -2 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{89} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{90} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{91} + ( 2 - 2 \beta_{2} ) q^{92} -2 \beta_{1} q^{93} + ( -1 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{94} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{95} -\beta_{1} q^{96} + ( -7 + 4 \beta_{1} + 3 \beta_{2} - 8 \beta_{3} ) q^{97} - q^{98} + ( -3 \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} + q^{3} - 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 4q^{8} + 5q^{9} + O(q^{10})$$ $$4q + 2q^{2} + q^{3} - 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 4q^{8} + 5q^{9} + 4q^{10} + 2q^{11} + q^{12} + q^{13} - 2q^{14} - q^{15} - 2q^{16} + 12q^{17} - 5q^{18} + 6q^{19} + 2q^{20} + 2q^{21} - 2q^{22} + 4q^{23} - q^{24} - 2q^{25} + 2q^{26} + 16q^{27} - 4q^{28} + 4q^{29} + q^{30} + 4q^{31} + 2q^{32} + 2q^{33} + 6q^{34} + 4q^{35} - 10q^{36} + 20q^{37} + 3q^{38} - 17q^{39} - 2q^{40} + 7q^{41} + q^{42} - 13q^{43} - 4q^{44} + 10q^{45} + 8q^{46} + 7q^{47} - 2q^{48} - 2q^{49} + 2q^{50} + 3q^{51} + q^{52} - 20q^{53} + 8q^{54} + 4q^{55} - 2q^{56} - 15q^{57} - 4q^{58} - 5q^{59} + 2q^{60} + 8q^{62} - 5q^{63} + 4q^{64} - q^{65} + q^{66} - 3q^{67} - 6q^{68} - 2q^{69} + 2q^{70} - 30q^{71} - 5q^{72} + 12q^{73} + 10q^{74} - 2q^{75} - 3q^{76} - 2q^{77} - 16q^{78} - 17q^{79} - 4q^{80} - 7q^{81} + 14q^{82} - 7q^{83} - q^{84} + 6q^{85} + 13q^{86} + 4q^{87} - 2q^{88} - 16q^{89} + 5q^{90} + 2q^{91} + 4q^{92} - 2q^{93} - 7q^{94} + 3q^{95} - q^{96} - 10q^{97} - 4q^{98} - 5q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + 2 \beta_{1} + 3$$

## Character Values

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

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$ $$1$$

## 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
 1.68614 − 0.396143i −1.18614 + 1.26217i 1.68614 + 0.396143i −1.18614 − 1.26217i
0.500000 + 0.866025i −1.18614 1.26217i −0.500000 + 0.866025i 0.500000 0.866025i 0.500000 1.65831i 0.500000 + 0.866025i −1.00000 −0.186141 + 2.99422i 1.00000
211.2 0.500000 + 0.866025i 1.68614 + 0.396143i −0.500000 + 0.866025i 0.500000 0.866025i 0.500000 + 1.65831i 0.500000 + 0.866025i −1.00000 2.68614 + 1.33591i 1.00000
421.1 0.500000 0.866025i −1.18614 + 1.26217i −0.500000 0.866025i 0.500000 + 0.866025i 0.500000 + 1.65831i 0.500000 0.866025i −1.00000 −0.186141 2.99422i 1.00000
421.2 0.500000 0.866025i 1.68614 0.396143i −0.500000 0.866025i 0.500000 + 0.866025i 0.500000 1.65831i 0.500000 0.866025i −1.00000 2.68614 1.33591i 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 $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{2} - T_{11} + 1$$ $$T_{13}^{4} - T_{13}^{3} + 9 T_{13}^{2} + 8 T_{13} + 64$$