# Properties

 Label 670.2.e.h 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: 6.0.2696112.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 5 x^{4} + 18 x^{2} - 8 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{5} + 15 \nu^{4} + 7 \nu^{3} + 54 \nu^{2} - 24 \nu + 202$$$$)/82$$ $$\beta_{2}$$ $$=$$ $$($$$$5 \nu^{5} - 25 \nu^{4} + 43 \nu^{3} - 90 \nu^{2} + 40 \nu - 282$$$$)/82$$ $$\beta_{3}$$ $$=$$ $$($$$$-10 \nu^{5} + 9 \nu^{4} - 45 \nu^{3} - 25 \nu^{2} - 162 \nu + 72$$$$)/82$$ $$\beta_{4}$$ $$=$$ $$($$$$-26 \nu^{5} + 7 \nu^{4} - 117 \nu^{3} - 65 \nu^{2} - 536 \nu - 26$$$$)/82$$ $$\beta_{5}$$ $$=$$ $$($$$$-26 \nu^{5} + 7 \nu^{4} - 117 \nu^{3} - 65 \nu^{2} - 372 \nu - 26$$$$)/82$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - 6 \beta_{3} - \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{2} + 5 \beta_{1} - 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-7 \beta_{5} - 3 \beta_{4} + 26 \beta_{3} - 26$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-25 \beta_{5} + 11 \beta_{4} + 22 \beta_{3} - 11 \beta_{2} - 25 \beta_{1}$$$$)/2$$

## Character Values

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

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

## 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
 −0.906803 + 1.57063i 1.17146 − 2.02903i 0.235342 − 0.407624i −0.906803 − 1.57063i 1.17146 + 2.02903i 0.235342 + 0.407624i
0.500000 + 0.866025i −2.10278 −0.500000 + 0.866025i 1.00000 −1.05139 1.82106i 1.55139 2.68708i −1.00000 1.42166 0.500000 + 0.866025i
171.2 0.500000 + 0.866025i −0.146365 −0.500000 + 0.866025i 1.00000 −0.0731827 0.126756i 0.573183 0.992782i −1.00000 −2.97858 0.500000 + 0.866025i
171.3 0.500000 + 0.866025i 3.24914 −0.500000 + 0.866025i 1.00000 1.62457 + 2.81384i −1.12457 + 1.94781i −1.00000 7.55691 0.500000 + 0.866025i
431.1 0.500000 0.866025i −2.10278 −0.500000 0.866025i 1.00000 −1.05139 + 1.82106i 1.55139 + 2.68708i −1.00000 1.42166 0.500000 0.866025i
431.2 0.500000 0.866025i −0.146365 −0.500000 0.866025i 1.00000 −0.0731827 + 0.126756i 0.573183 + 0.992782i −1.00000 −2.97858 0.500000 0.866025i
431.3 0.500000 0.866025i 3.24914 −0.500000 0.866025i 1.00000 1.62457 2.81384i −1.12457 1.94781i −1.00000 7.55691 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 $$S_{2}^{\mathrm{new}}(670, [\chi])$$:

 $$T_{3}^{3} - T_{3}^{2} - 7 T_{3} - 1$$ $$T_{7}^{6} - 2 T_{7}^{5} + 10 T_{7}^{4} - 4 T_{7}^{3} + 52 T_{7}^{2} - 48 T_{7} + 64$$ $$T_{11}^{2} + 4 T_{11} + 16$$