# Properties

 Label 670.2.e.f Level 670 Weight 2 Character orbit 670.e Analytic conductor 5.350 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$670 = 2 \cdot 5 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 670.e (of order $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$5.34997693543$$ 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: $$3$$ 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.

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

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

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

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/670\mathbb{Z}\right)^\times$$.

 $$n$$ $$471$$ $$537$$ $$\chi(n)$$ $$-1 + \beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
171.1
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
−0.500000 0.866025i −1.44949 −0.500000 + 0.866025i −1.00000 0.724745 + 1.25529i 0.224745 0.389270i 1.00000 −0.898979 0.500000 + 0.866025i
171.2 −0.500000 0.866025i 3.44949 −0.500000 + 0.866025i −1.00000 −1.72474 2.98735i −2.22474 + 3.85337i 1.00000 8.89898 0.500000 + 0.866025i
431.1 −0.500000 + 0.866025i −1.44949 −0.500000 0.866025i −1.00000 0.724745 1.25529i 0.224745 + 0.389270i 1.00000 −0.898979 0.500000 0.866025i
431.2 −0.500000 + 0.866025i 3.44949 −0.500000 0.866025i −1.00000 −1.72474 + 2.98735i −2.22474 3.85337i 1.00000 8.89898 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
67.c Even 1 yes

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(670, [\chi])$$:

 $$T_{3}^{2} - 2 T_{3} - 5$$ $$T_{7}^{4} + 4 T_{7}^{3} + 18 T_{7}^{2} - 8 T_{7} + 4$$ $$T_{11}$$