# Properties

 Label 630.2.i.e Level 630 Weight 2 Character orbit 630.i 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.i (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: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{3} + 2 \beta_{2}$$$$)/3$$

## Character Values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
1.00000 −1.50000 + 0.866025i 1.00000 0.500000 + 0.866025i −1.50000 + 0.866025i −2.62132 + 0.358719i 1.00000 1.50000 2.59808i 0.500000 + 0.866025i
121.2 1.00000 −1.50000 + 0.866025i 1.00000 0.500000 + 0.866025i −1.50000 + 0.866025i 1.62132 2.09077i 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 −2.62132 0.358719i 1.00000 1.50000 + 2.59808i 0.500000 0.866025i
151.2 1.00000 −1.50000 0.866025i 1.00000 0.500000 0.866025i −1.50000 0.866025i 1.62132 + 2.09077i 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 $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{4} + 18 T_{11}^{2} + 324$$ $$T_{13}^{2} + 2 T_{13} + 4$$