# Properties

 Label 670.2.e.g Level 670 Weight 2 Character orbit 670.e Analytic conductor 5.350 Analytic rank 0 Dimension 6 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: $$6$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu^{4} - \nu^{3} + 9 \nu^{2} - 21 \nu - 9$$$$)/27$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu^{4} - \nu^{3} - 18 \nu^{2} + 33 \nu - 9$$$$)/27$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{5} - \nu^{4} - 2 \nu^{3} + 12 \nu + 36$$$$)/27$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} - \nu^{4} + 7 \nu^{3} + 9 \nu^{2} + 12 \nu + 9$$$$)/27$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} + 18 \nu^{2} + 3 \nu - 72$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_{1} + 2$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{5} + 7 \beta_{4} - 11 \beta_{3} + \beta_{2} - \beta_{1} + 7$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 8 \beta_{2} + 10 \beta_{1} + 20$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$7 \beta_{5} - 2 \beta_{4} - 20 \beta_{3} + \beta_{2} - 10 \beta_{1} + 43$$$$)/3$$

## Character Values

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

 $$n$$ $$471$$ $$537$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$

## 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
 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 −2.51414 −0.500000 + 0.866025i 1.00000 −1.25707 2.17731i −0.0966262 + 0.167362i −1.00000 3.32088 0.500000 + 0.866025i
171.2 0.500000 + 0.866025i −0.571993 −0.500000 + 0.866025i 1.00000 −0.285997 0.495361i −2.12241 + 3.67612i −1.00000 −2.67282 0.500000 + 0.866025i
171.3 0.500000 + 0.866025i 2.08613 −0.500000 + 0.866025i 1.00000 1.04307 + 1.80664i 1.21903 2.11143i −1.00000 1.35194 0.500000 + 0.866025i
431.1 0.500000 0.866025i −2.51414 −0.500000 0.866025i 1.00000 −1.25707 + 2.17731i −0.0966262 0.167362i −1.00000 3.32088 0.500000 0.866025i
431.2 0.500000 0.866025i −0.571993 −0.500000 0.866025i 1.00000 −0.285997 + 0.495361i −2.12241 3.67612i −1.00000 −2.67282 0.500000 0.866025i
431.3 0.500000 0.866025i 2.08613 −0.500000 0.866025i 1.00000 1.04307 1.80664i 1.21903 + 2.11143i −1.00000 1.35194 0.500000 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 431.3 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
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 $$S_{2}^{\mathrm{new}}(670, [\chi])$$:

 $$T_{3}^{3} + T_{3}^{2} - 5 T_{3} - 3$$ $$T_{7}^{6} + 2 T_{7}^{5} + 14 T_{7}^{4} - 16 T_{7}^{3} + 104 T_{7}^{2} + 20 T_{7} + 4$$ $$T_{11}^{6} - 8 T_{11}^{5} + 52 T_{11}^{4} - 120 T_{11}^{3} + 240 T_{11}^{2} + 144 T_{11} + 144$$