# Properties

 Label 670.2.e.i Level 670 Weight 2 Character orbit 670.e Analytic conductor 5.350 Analytic rank 0 Dimension 8 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 31 x^{6} - 2 x^{5} + 597 x^{4} - 4 x^{3} + 5860 x^{2} + 5264 x + 35344$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-187961 \nu^{7} + 1128439 \nu^{6} - 5753189 \nu^{5} + 16903425 \nu^{4} - 79046227 \nu^{3} + 519581413 \nu^{2} - 628090396 \nu + 4413343428$$$$)/ 4124271620$$ $$\beta_{3}$$ $$=$$ $$($$$$-81 \nu^{7} + 674 \nu^{6} - 1779 \nu^{5} - 3570 \nu^{4} + 4163 \nu^{3} - 50952 \nu^{2} - 137616 \nu - 2402152$$$$)/808760$$ $$\beta_{4}$$ $$=$$ $$($$$$526409 \nu^{7} - 4498811 \nu^{6} + 22936561 \nu^{5} - 153112435 \nu^{4} + 315138023 \nu^{3} - 2071444337 \nu^{2} + 993474914 \nu - 17594923572$$$$)/ 4124271620$$ $$\beta_{5}$$ $$=$$ $$($$$$16011 \nu^{7} + 1566 \nu^{6} + 351649 \nu^{5} + 705670 \nu^{4} + 11038927 \nu^{3} + 10071512 \nu^{2} + 27202096 \nu + 229097132$$$$)/87750460$$ $$\beta_{6}$$ $$=$$ $$($$$$128 \nu^{7} + 183 \nu^{6} - 933 \nu^{5} + 13130 \nu^{4} - 12819 \nu^{3} + 84261 \nu^{2} - 493942 \nu + 715716$$$$)/202190$$ $$\beta_{7}$$ $$=$$ $$($$$$40419 \nu^{7} - 134136 \nu^{6} + 1696481 \nu^{5} - 1251420 \nu^{4} + 21781083 \nu^{3} + 2577578 \nu^{2} + 203328924 \nu + 122214288$$$$)/38011720$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 14 \beta_{2} + \beta_{1} - 15$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{7} - 2 \beta_{6} + 15 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 29$$ $$\nu^{4}$$ $$=$$ $$-4 \beta_{6} - 58 \beta_{4} - 218 \beta_{2} - 29 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$62 \beta_{7} - 247 \beta_{5} - 176 \beta_{4} + 176 \beta_{3} - 422 \beta_{2} - 247 \beta_{1} + 669$$ $$\nu^{6}$$ $$=$$ $$176 \beta_{7} + 176 \beta_{6} - 669 \beta_{5} - 176 \beta_{4} + 1538 \beta_{3} + 4375$$ $$\nu^{7}$$ $$=$$ $$1538 \beta_{6} + 3802 \beta_{4} + 10070 \beta_{2} + 4375 \beta_{1}$$

## Character Values

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

 $$n$$ $$471$$ $$537$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$

## 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
 1.93979 − 3.35982i −1.43979 + 2.49379i −1.81630 + 3.14593i 2.31630 − 4.01195i 1.93979 + 3.35982i −1.43979 − 2.49379i −1.81630 − 3.14593i 2.31630 + 4.01195i
−0.500000 0.866025i −0.414214 −0.500000 + 0.866025i 1.00000 0.207107 + 0.358719i −0.707107 + 1.22474i 1.00000 −2.82843 −0.500000 0.866025i
171.2 −0.500000 0.866025i −0.414214 −0.500000 + 0.866025i 1.00000 0.207107 + 0.358719i −0.707107 + 1.22474i 1.00000 −2.82843 −0.500000 0.866025i
171.3 −0.500000 0.866025i 2.41421 −0.500000 + 0.866025i 1.00000 −1.20711 2.09077i 0.707107 1.22474i 1.00000 2.82843 −0.500000 0.866025i
171.4 −0.500000 0.866025i 2.41421 −0.500000 + 0.866025i 1.00000 −1.20711 2.09077i 0.707107 1.22474i 1.00000 2.82843 −0.500000 0.866025i
431.1 −0.500000 + 0.866025i −0.414214 −0.500000 0.866025i 1.00000 0.207107 0.358719i −0.707107 1.22474i 1.00000 −2.82843 −0.500000 + 0.866025i
431.2 −0.500000 + 0.866025i −0.414214 −0.500000 0.866025i 1.00000 0.207107 0.358719i −0.707107 1.22474i 1.00000 −2.82843 −0.500000 + 0.866025i
431.3 −0.500000 + 0.866025i 2.41421 −0.500000 0.866025i 1.00000 −1.20711 + 2.09077i 0.707107 + 1.22474i 1.00000 2.82843 −0.500000 + 0.866025i
431.4 −0.500000 + 0.866025i 2.41421 −0.500000 0.866025i 1.00000 −1.20711 + 2.09077i 0.707107 + 1.22474i 1.00000 2.82843 −0.500000 + 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 431.4 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 $$S_{2}^{\mathrm{new}}(670, [\chi])$$:

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