# Properties

 Label 630.2.j.l Level 630 Weight 2 Character orbit 630.j Analytic conductor 5.031 Analytic rank 0 Dimension 8 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.856615824.2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} - 7 \nu^{3} - 10 \nu + 2$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} - \nu^{3} + 6 \nu^{2} - 2 \nu + 4$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{4} + 2 \nu^{3} - 2 \nu^{2} + 10 \nu - 8$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + \nu^{5} + 9 \nu^{4} + 5 \nu^{3} + 22 \nu^{2} + 10$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 10 \nu^{5} + 4 \nu^{4} + 31 \nu^{3} + 22 \nu^{2} + 30 \nu + 10$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + 10 \nu^{5} - 9 \nu^{4} + 29 \nu^{3} - 20 \nu^{2} + 20 \nu - 6$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} - 11 \nu^{5} + 9 \nu^{4} - 32 \nu^{3} + 24 \nu^{2} - 10 \nu + 16$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{7} + 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} - 8$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 10 \beta_{2} + 2 \beta_{1} - 3$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-11 \beta_{7} - 18 \beta_{6} + 7 \beta_{5} - 11 \beta_{4} + 4 \beta_{3} + 35$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-7 \beta_{7} - 7 \beta_{5} - 11 \beta_{4} + 18 \beta_{3} + 50 \beta_{2} - 16 \beta_{1} + 17$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$19 \beta_{7} + 32 \beta_{6} - 13 \beta_{5} + 21 \beta_{4} - 4 \beta_{3} + 2 \beta_{1} - 57$$ $$\nu^{7}$$ $$=$$ $$($$$$39 \beta_{7} + 6 \beta_{6} + 45 \beta_{5} + 47 \beta_{4} - 80 \beta_{3} - 250 \beta_{2} + 128 \beta_{1} - 101$$$$)/3$$

## Character Values

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

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

## 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
 − 2.06288i 0.385731i 2.33086i 1.07834i 2.06288i − 0.385731i − 2.33086i − 1.07834i
−0.500000 0.866025i −1.70236 0.319344i −0.500000 + 0.866025i 0.500000 0.866025i 0.574618 + 1.63396i 0.500000 + 0.866025i 1.00000 2.79604 + 1.08728i −1.00000
211.2 −0.500000 0.866025i −0.805046 + 1.53359i −0.500000 + 0.866025i 0.500000 0.866025i 1.73065 0.0696054i 0.500000 + 0.866025i 1.00000 −1.70380 2.46922i −1.00000
211.3 −0.500000 0.866025i −0.657430 1.60243i −0.500000 + 0.866025i 0.500000 0.866025i −1.05903 + 1.37057i 0.500000 + 0.866025i 1.00000 −2.13557 + 2.10697i −1.00000
211.4 −0.500000 0.866025i 1.66483 0.477841i −0.500000 + 0.866025i 0.500000 0.866025i −1.24624 1.20287i 0.500000 + 0.866025i 1.00000 2.54334 1.59105i −1.00000
421.1 −0.500000 + 0.866025i −1.70236 + 0.319344i −0.500000 0.866025i 0.500000 + 0.866025i 0.574618 1.63396i 0.500000 0.866025i 1.00000 2.79604 1.08728i −1.00000
421.2 −0.500000 + 0.866025i −0.805046 1.53359i −0.500000 0.866025i 0.500000 + 0.866025i 1.73065 + 0.0696054i 0.500000 0.866025i 1.00000 −1.70380 + 2.46922i −1.00000
421.3 −0.500000 + 0.866025i −0.657430 + 1.60243i −0.500000 0.866025i 0.500000 + 0.866025i −1.05903 1.37057i 0.500000 0.866025i 1.00000 −2.13557 2.10697i −1.00000
421.4 −0.500000 + 0.866025i 1.66483 + 0.477841i −0.500000 0.866025i 0.500000 + 0.866025i −1.24624 + 1.20287i 0.500000 0.866025i 1.00000 2.54334 + 1.59105i −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 421.4 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 $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{8} - \cdots$$ $$T_{13}^{8} - \cdots$$