from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1021, base_ring=CyclotomicField(510))
M = H._module
chi = DirichletCharacter(H, M([344]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,1021))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1021\) | |
| Conductor: | \(1021\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(255\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{255})$ |
| Fixed field: | Number field defined by a degree 255 polynomial (not computed) |
First 31 of 128 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1021}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{85}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{19}{85}\right)\) | \(e\left(\frac{16}{255}\right)\) | \(e\left(\frac{62}{85}\right)\) | \(e\left(\frac{77}{85}\right)\) | \(e\left(\frac{71}{85}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{172}{255}\right)\) | \(e\left(\frac{202}{255}\right)\) |
| \(\chi_{1021}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{85}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{38}{85}\right)\) | \(e\left(\frac{202}{255}\right)\) | \(e\left(\frac{39}{85}\right)\) | \(e\left(\frac{69}{85}\right)\) | \(e\left(\frac{57}{85}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{4}{255}\right)\) | \(e\left(\frac{64}{255}\right)\) |
| \(\chi_{1021}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{85}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{24}{85}\right)\) | \(e\left(\frac{56}{255}\right)\) | \(e\left(\frac{47}{85}\right)\) | \(e\left(\frac{57}{85}\right)\) | \(e\left(\frac{36}{85}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{92}{255}\right)\) | \(e\left(\frac{197}{255}\right)\) |
| \(\chi_{1021}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{85}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{38}{85}\right)\) | \(e\left(\frac{32}{255}\right)\) | \(e\left(\frac{39}{85}\right)\) | \(e\left(\frac{69}{85}\right)\) | \(e\left(\frac{57}{85}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{89}{255}\right)\) | \(e\left(\frac{149}{255}\right)\) |
| \(\chi_{1021}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{85}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{69}{85}\right)\) | \(e\left(\frac{76}{255}\right)\) | \(e\left(\frac{82}{85}\right)\) | \(e\left(\frac{47}{85}\right)\) | \(e\left(\frac{61}{85}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{52}{255}\right)\) | \(e\left(\frac{67}{255}\right)\) |
| \(\chi_{1021}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{85}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{8}{85}\right)\) | \(e\left(\frac{47}{255}\right)\) | \(e\left(\frac{44}{85}\right)\) | \(e\left(\frac{19}{85}\right)\) | \(e\left(\frac{12}{85}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{59}{255}\right)\) | \(e\left(\frac{179}{255}\right)\) |
| \(\chi_{1021}(51,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{85}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{78}{85}\right)\) | \(e\left(\frac{182}{255}\right)\) | \(e\left(\frac{4}{85}\right)\) | \(e\left(\frac{79}{85}\right)\) | \(e\left(\frac{32}{85}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{44}{255}\right)\) | \(e\left(\frac{194}{255}\right)\) |
| \(\chi_{1021}(55,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{85}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{57}{85}\right)\) | \(e\left(\frac{218}{255}\right)\) | \(e\left(\frac{16}{85}\right)\) | \(e\left(\frac{61}{85}\right)\) | \(e\left(\frac{43}{85}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{176}{255}\right)\) | \(e\left(\frac{11}{255}\right)\) |
| \(\chi_{1021}(60,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{85}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{32}{85}\right)\) | \(e\left(\frac{103}{255}\right)\) | \(e\left(\frac{6}{85}\right)\) | \(e\left(\frac{76}{85}\right)\) | \(e\left(\frac{48}{85}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{151}{255}\right)\) | \(e\left(\frac{121}{255}\right)\) |
| \(\chi_{1021}(68,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{85}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{41}{85}\right)\) | \(e\left(\frac{209}{255}\right)\) | \(e\left(\frac{13}{85}\right)\) | \(e\left(\frac{23}{85}\right)\) | \(e\left(\frac{19}{85}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{143}{255}\right)\) | \(e\left(\frac{248}{255}\right)\) |
| \(\chi_{1021}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{56}{85}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{27}{85}\right)\) | \(e\left(\frac{148}{255}\right)\) | \(e\left(\frac{21}{85}\right)\) | \(e\left(\frac{11}{85}\right)\) | \(e\left(\frac{83}{85}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{61}{255}\right)\) | \(e\left(\frac{211}{255}\right)\) |
| \(\chi_{1021}(87,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{48}{85}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{11}{85}\right)\) | \(e\left(\frac{224}{255}\right)\) | \(e\left(\frac{18}{85}\right)\) | \(e\left(\frac{58}{85}\right)\) | \(e\left(\frac{59}{85}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{113}{255}\right)\) | \(e\left(\frac{23}{255}\right)\) |
| \(\chi_{1021}(99,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{85}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{3}{85}\right)\) | \(e\left(\frac{7}{255}\right)\) | \(e\left(\frac{59}{85}\right)\) | \(e\left(\frac{39}{85}\right)\) | \(e\left(\frac{47}{85}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{139}{255}\right)\) | \(e\left(\frac{184}{255}\right)\) |
| \(\chi_{1021}(114,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{85}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{54}{85}\right)\) | \(e\left(\frac{211}{255}\right)\) | \(e\left(\frac{42}{85}\right)\) | \(e\left(\frac{22}{85}\right)\) | \(e\left(\frac{81}{85}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{37}{255}\right)\) | \(e\left(\frac{82}{255}\right)\) |
| \(\chi_{1021}(116,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{72}{85}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{59}{85}\right)\) | \(e\left(\frac{251}{255}\right)\) | \(e\left(\frac{27}{85}\right)\) | \(e\left(\frac{2}{85}\right)\) | \(e\left(\frac{46}{85}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{212}{255}\right)\) | \(e\left(\frac{77}{255}\right)\) |
| \(\chi_{1021}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{85}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{76}{85}\right)\) | \(e\left(\frac{149}{255}\right)\) | \(e\left(\frac{78}{85}\right)\) | \(e\left(\frac{53}{85}\right)\) | \(e\left(\frac{29}{85}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{8}{255}\right)\) | \(e\left(\frac{128}{255}\right)\) |
| \(\chi_{1021}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{85}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{49}{85}\right)\) | \(e\left(\frac{1}{255}\right)\) | \(e\left(\frac{57}{85}\right)\) | \(e\left(\frac{42}{85}\right)\) | \(e\left(\frac{31}{85}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{202}{255}\right)\) | \(e\left(\frac{172}{255}\right)\) |
| \(\chi_{1021}(154,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{85}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{46}{85}\right)\) | \(e\left(\frac{79}{255}\right)\) | \(e\left(\frac{83}{85}\right)\) | \(e\left(\frac{3}{85}\right)\) | \(e\left(\frac{69}{85}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{148}{255}\right)\) | \(e\left(\frac{73}{255}\right)\) |
| \(\chi_{1021}(164,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{85}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{23}{85}\right)\) | \(e\left(\frac{82}{255}\right)\) | \(e\left(\frac{84}{85}\right)\) | \(e\left(\frac{44}{85}\right)\) | \(e\left(\frac{77}{85}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{244}{255}\right)\) | \(e\left(\frac{79}{255}\right)\) |
| \(\chi_{1021}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{85}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{16}{85}\right)\) | \(e\left(\frac{94}{255}\right)\) | \(e\left(\frac{3}{85}\right)\) | \(e\left(\frac{38}{85}\right)\) | \(e\left(\frac{24}{85}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{118}{255}\right)\) | \(e\left(\frac{103}{255}\right)\) |
| \(\chi_{1021}(176,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{85}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{14}{85}\right)\) | \(e\left(\frac{61}{255}\right)\) | \(e\left(\frac{77}{85}\right)\) | \(e\left(\frac{12}{85}\right)\) | \(e\left(\frac{21}{85}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{82}{255}\right)\) | \(e\left(\frac{37}{255}\right)\) |
| \(\chi_{1021}(183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{85}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{33}{85}\right)\) | \(e\left(\frac{247}{255}\right)\) | \(e\left(\frac{54}{85}\right)\) | \(e\left(\frac{4}{85}\right)\) | \(e\left(\frac{7}{85}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{169}{255}\right)\) | \(e\left(\frac{154}{255}\right)\) |
| \(\chi_{1021}(193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{85}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{62}{85}\right)\) | \(e\left(\frac{173}{255}\right)\) | \(e\left(\frac{1}{85}\right)\) | \(e\left(\frac{41}{85}\right)\) | \(e\left(\frac{8}{85}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{11}{255}\right)\) | \(e\left(\frac{176}{255}\right)\) |
| \(\chi_{1021}(197,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{85}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{19}{85}\right)\) | \(e\left(\frac{101}{255}\right)\) | \(e\left(\frac{62}{85}\right)\) | \(e\left(\frac{77}{85}\right)\) | \(e\left(\frac{71}{85}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{2}{255}\right)\) | \(e\left(\frac{32}{255}\right)\) |
| \(\chi_{1021}(206,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{85}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{31}{85}\right)\) | \(e\left(\frac{44}{255}\right)\) | \(e\left(\frac{43}{85}\right)\) | \(e\left(\frac{63}{85}\right)\) | \(e\left(\frac{4}{85}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{218}{255}\right)\) | \(e\left(\frac{173}{255}\right)\) |
| \(\chi_{1021}(207,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{85}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{74}{85}\right)\) | \(e\left(\frac{116}{255}\right)\) | \(e\left(\frac{67}{85}\right)\) | \(e\left(\frac{27}{85}\right)\) | \(e\left(\frac{26}{85}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{227}{255}\right)\) | \(e\left(\frac{62}{255}\right)\) |
| \(\chi_{1021}(222,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{85}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{61}{85}\right)\) | \(e\left(\frac{199}{255}\right)\) | \(e\left(\frac{38}{85}\right)\) | \(e\left(\frac{28}{85}\right)\) | \(e\left(\frac{49}{85}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{163}{255}\right)\) | \(e\left(\frac{58}{255}\right)\) |
| \(\chi_{1021}(225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{85}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{3}{85}\right)\) | \(e\left(\frac{92}{255}\right)\) | \(e\left(\frac{59}{85}\right)\) | \(e\left(\frac{39}{85}\right)\) | \(e\left(\frac{47}{85}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{224}{255}\right)\) | \(e\left(\frac{14}{255}\right)\) |
| \(\chi_{1021}(233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{85}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{7}{85}\right)\) | \(e\left(\frac{158}{255}\right)\) | \(e\left(\frac{81}{85}\right)\) | \(e\left(\frac{6}{85}\right)\) | \(e\left(\frac{53}{85}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{41}{255}\right)\) | \(e\left(\frac{146}{255}\right)\) |
| \(\chi_{1021}(244,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{85}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{81}{85}\right)\) | \(e\left(\frac{19}{255}\right)\) | \(e\left(\frac{63}{85}\right)\) | \(e\left(\frac{33}{85}\right)\) | \(e\left(\frac{79}{85}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{13}{255}\right)\) | \(e\left(\frac{208}{255}\right)\) |
| \(\chi_{1021}(254,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{85}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{31}{85}\right)\) | \(e\left(\frac{214}{255}\right)\) | \(e\left(\frac{43}{85}\right)\) | \(e\left(\frac{63}{85}\right)\) | \(e\left(\frac{4}{85}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{133}{255}\right)\) | \(e\left(\frac{88}{255}\right)\) |