from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(388080, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([0,210,0,315,370,294]))
chi.galois_orbit()
[g,chi] = znchar(Mod(73,388080))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(388080\) | |
| Conductor: | \(21560\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(420\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 21560.om | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{420})$ |
| Fixed field: | Number field defined by a degree 420 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{388080}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{140}\right)\) | \(e\left(\frac{31}{420}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{353}{420}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{317}{420}\right)\) |
| \(\chi_{388080}(1657,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{140}\right)\) | \(e\left(\frac{101}{420}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{283}{420}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{247}{420}\right)\) |
| \(\chi_{388080}(3097,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{140}\right)\) | \(e\left(\frac{37}{420}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{191}{420}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{419}{420}\right)\) |
| \(\chi_{388080}(8137,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{140}\right)\) | \(e\left(\frac{229}{420}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{47}{420}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{323}{420}\right)\) |
| \(\chi_{388080}(15193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{140}\right)\) | \(e\left(\frac{103}{420}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{89}{420}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{281}{420}\right)\) |
| \(\chi_{388080}(25273,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{81}{140}\right)\) | \(e\left(\frac{67}{420}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{221}{420}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{89}{420}\right)\) |
| \(\chi_{388080}(26857,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{140}\right)\) | \(e\left(\frac{257}{420}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{271}{420}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{379}{420}\right)\) |
| \(\chi_{388080}(31897,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{140}\right)\) | \(e\left(\frac{389}{420}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{67}{420}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{103}{420}\right)\) |
| \(\chi_{388080}(33337,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{140}\right)\) | \(e\left(\frac{181}{420}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{83}{420}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{347}{420}\right)\) |
| \(\chi_{388080}(38953,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{140}\right)\) | \(e\left(\frac{263}{420}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{109}{420}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{61}{420}\right)\) |
| \(\chi_{388080}(48457,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{99}{140}\right)\) | \(e\left(\frac{253}{420}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{239}{420}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{311}{420}\right)\) |
| \(\chi_{388080}(49033,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{107}{420}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{121}{420}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{349}{420}\right)\) |
| \(\chi_{388080}(54073,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{140}\right)\) | \(e\left(\frac{239}{420}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{337}{420}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{73}{420}\right)\) |
| \(\chi_{388080}(55513,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{211}{420}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{113}{420}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{17}{420}\right)\) |
| \(\chi_{388080}(57097,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{140}\right)\) | \(e\left(\frac{41}{420}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{223}{420}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{67}{420}\right)\) |
| \(\chi_{388080}(63577,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{87}{140}\right)\) | \(e\left(\frac{409}{420}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{227}{420}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{23}{420}\right)\) |
| \(\chi_{388080}(70633,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{129}{140}\right)\) | \(e\left(\frac{283}{420}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{269}{420}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{401}{420}\right)\) |
| \(\chi_{388080}(72217,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{140}\right)\) | \(e\left(\frac{353}{420}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{199}{420}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{331}{420}\right)\) |
| \(\chi_{388080}(79273,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{140}\right)\) | \(e\left(\frac{311}{420}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{73}{420}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{37}{420}\right)\) |
| \(\chi_{388080}(80713,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{247}{420}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{401}{420}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{209}{420}\right)\) |
| \(\chi_{388080}(82297,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{140}\right)\) | \(e\left(\frac{197}{420}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{211}{420}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{199}{420}\right)\) |
| \(\chi_{388080}(85753,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{117}{140}\right)\) | \(e\left(\frac{19}{420}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{257}{420}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{113}{420}\right)\) |
| \(\chi_{388080}(88777,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{123}{140}\right)\) | \(e\left(\frac{361}{420}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{263}{420}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{47}{420}\right)\) |
| \(\chi_{388080}(103897,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{140}\right)\) | \(e\left(\frac{13}{420}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{419}{420}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{11}{420}\right)\) |
| \(\chi_{388080}(104473,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{140}\right)\) | \(e\left(\frac{47}{420}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{61}{420}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{169}{420}\right)\) |
| \(\chi_{388080}(109513,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{140}\right)\) | \(e\left(\frac{179}{420}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{277}{420}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{313}{420}\right)\) |
| \(\chi_{388080}(110953,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{140}\right)\) | \(e\left(\frac{391}{420}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{293}{420}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{137}{420}\right)\) |
| \(\chi_{388080}(112537,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{140}\right)\) | \(e\left(\frac{401}{420}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{163}{420}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{307}{420}\right)\) |
| \(\chi_{388080}(113977,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{140}\right)\) | \(e\left(\frac{397}{420}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{131}{420}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{239}{420}\right)\) |
| \(\chi_{388080}(119017,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{140}\right)\) | \(e\left(\frac{169}{420}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{407}{420}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{143}{420}\right)\) |
| \(\chi_{388080}(126073,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{140}\right)\) | \(e\left(\frac{43}{420}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{29}{420}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{101}{420}\right)\) |