# Properties

 Label 630.2.j.h 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ 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 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## 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
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
0.500000 + 0.866025i −0.724745 + 1.57313i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.72474 + 0.158919i −0.500000 0.866025i −1.00000 −1.94949 2.28024i −1.00000
211.2 0.500000 + 0.866025i 1.72474 + 0.158919i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.724745 + 1.57313i −0.500000 0.866025i −1.00000 2.94949 + 0.548188i −1.00000
421.1 0.500000 0.866025i −0.724745 1.57313i −0.500000 0.866025i −0.500000 0.866025i −1.72474 0.158919i −0.500000 + 0.866025i −1.00000 −1.94949 + 2.28024i −1.00000
421.2 0.500000 0.866025i 1.72474 0.158919i −0.500000 0.866025i −0.500000 0.866025i 0.724745 1.57313i −0.500000 + 0.866025i −1.00000 2.94949 0.548188i −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 $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{4} - 6 T_{11}^{3} + 33 T_{11}^{2} - 18 T_{11} + 9$$ $$T_{13}^{4} + 4 T_{13}^{3} + 18 T_{13}^{2} - 8 T_{13} + 4$$