# Properties

 Label 630.2.j.j Level 630 Weight 2 Character orbit 630.j Analytic conductor 5.031 Analytic rank 0 Dimension 6 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.954288.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} - 5 \nu^{4} - \nu^{3} - 9 \nu^{2} + 6 \nu + 45$$$$)/27$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{5} + \nu^{4} + 2 \nu^{3} - 12 \nu - 36$$$$)/27$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} + \nu^{3} + 6 \nu^{2} + 3 \nu - 9$$$$)/9$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} - 2 \nu^{3} + 3 \nu^{2} - 6 \nu - 9$$$$)/9$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{5} + 7 \nu^{4} - 13 \nu^{3} - 9 \nu^{2} - 3 \nu - 90$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_{1} - 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{5} + 2 \beta_{4} + \beta_{3} - 4 \beta_{2} - 4 \beta_{1} + 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} + 10 \beta_{2} + \beta_{1} + 2$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - \beta_{2} - 10 \beta_{1} + 16$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$7 \beta_{5} - 5 \beta_{4} + 11 \beta_{3} + 19 \beta_{2} + 10 \beta_{1} + 38$$$$)/3$$

## Character Values

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

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

## 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
 0.403374 + 1.68443i −1.62241 − 0.606458i 1.71903 − 0.211943i 0.403374 − 1.68443i −1.62241 + 0.606458i 1.71903 + 0.211943i
−0.500000 0.866025i −1.25707 1.19154i −0.500000 + 0.866025i 0.500000 0.866025i −0.403374 + 1.68443i −0.500000 0.866025i 1.00000 0.160442 + 2.99571i −1.00000
211.2 −0.500000 0.866025i −0.285997 + 1.70828i −0.500000 + 0.866025i 0.500000 0.866025i 1.62241 0.606458i −0.500000 0.866025i 1.00000 −2.83641 0.977122i −1.00000
211.3 −0.500000 0.866025i 1.04307 1.38276i −0.500000 + 0.866025i 0.500000 0.866025i −1.71903 0.211943i −0.500000 0.866025i 1.00000 −0.824030 2.88461i −1.00000
421.1 −0.500000 + 0.866025i −1.25707 + 1.19154i −0.500000 0.866025i 0.500000 + 0.866025i −0.403374 1.68443i −0.500000 + 0.866025i 1.00000 0.160442 2.99571i −1.00000
421.2 −0.500000 + 0.866025i −0.285997 1.70828i −0.500000 0.866025i 0.500000 + 0.866025i 1.62241 + 0.606458i −0.500000 + 0.866025i 1.00000 −2.83641 + 0.977122i −1.00000
421.3 −0.500000 + 0.866025i 1.04307 + 1.38276i −0.500000 0.866025i 0.500000 + 0.866025i −1.71903 + 0.211943i −0.500000 + 0.866025i 1.00000 −0.824030 + 2.88461i −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 421.3 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 $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{6} - 3 T_{11}^{5} + 24 T_{11}^{4} + 63 T_{11}^{3} + 198 T_{11}^{2} + 135 T_{11} + 81$$ $$T_{13}^{6} + 18 T_{13}^{4} - 52 T_{13}^{3} + 324 T_{13}^{2} - 468 T_{13} + 676$$