# Properties

 Label 670.2.e.d Level 670 Weight 2 Character orbit 670.e Analytic conductor 5.350 Analytic rank 0 Dimension 2 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 $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.34997693543$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$471$$ $$537$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −1.00000 0.500000 + 0.866025i −1.00000 + 1.73205i −1.00000 −2.00000 −0.500000 0.866025i
431.1 0.500000 0.866025i 1.00000 −0.500000 0.866025i −1.00000 0.500000 0.866025i −1.00000 1.73205i −1.00000 −2.00000 −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 Parity Ord Mult Type
1.a even 1 1 trivial
67.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 670.2.e.d 2
67.c even 3 1 inner 670.2.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.2.e.d 2 1.a even 1 1 trivial
670.2.e.d 2 67.c even 3 1 inner

## Hecke kernels

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

 $$T_{3} - 1$$ $$T_{7}^{2} + 2 T_{7} + 4$$ $$T_{11}^{2} + 6 T_{11} + 36$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$( 1 - T + 3 T^{2} )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$
$11$ $$1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$17$ $$1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4}$$
$19$ $$1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}$$
$29$ $$1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4}$$
$31$ $$1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4}$$
$37$ $$1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4}$$
$41$ $$1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{2}$$
$47$ $$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + 3 T + 53 T^{2} )^{2}$$
$59$ $$( 1 + 6 T + 59 T^{2} )^{2}$$
$61$ $$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$
$67$ $$1 - 5 T + 67 T^{2}$$
$71$ $$1 - 15 T + 154 T^{2} - 1065 T^{3} + 5041 T^{4}$$
$73$ $$1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4}$$
$79$ $$1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4}$$
$83$ $$1 - 15 T + 142 T^{2} - 1245 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 9 T + 89 T^{2} )^{2}$$
$97$ $$1 - 10 T + 3 T^{2} - 970 T^{3} + 9409 T^{4}$$