from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(869, base_ring=CyclotomicField(130))
M = H._module
chi = DirichletCharacter(H, M([117,35]))
chi.galois_orbit()
[g,chi] = znchar(Mod(17,869))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(869\) | |
Conductor: | \(869\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(130\) | 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: | $\Q(\zeta_{65})$ |
Fixed field: | Number field defined by a degree 130 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{869}(17,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{130}\right)\) | \(e\left(\frac{61}{130}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{121}{130}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) |
\(\chi_{869}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{130}\right)\) | \(e\left(\frac{47}{130}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) |
\(\chi_{869}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{130}\right)\) | \(e\left(\frac{29}{130}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{49}{130}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) |
\(\chi_{869}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{130}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) |
\(\chi_{869}(94,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{130}\right)\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{51}{130}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) |
\(\chi_{869}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{130}\right)\) | \(e\left(\frac{87}{130}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{17}{130}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) |
\(\chi_{869}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) |
\(\chi_{869}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{89}{130}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{119}{130}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) |
\(\chi_{869}(140,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{130}\right)\) | \(e\left(\frac{127}{130}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) |
\(\chi_{869}(150,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{130}\right)\) | \(e\left(\frac{33}{130}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{123}{130}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) |
\(\chi_{869}(172,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) |
\(\chi_{869}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{27}{130}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{77}{130}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) |
\(\chi_{869}(215,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{130}\right)\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) |
\(\chi_{869}(216,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{130}\right)\) | \(e\left(\frac{103}{130}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{53}{130}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) |
\(\chi_{869}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{130}\right)\) | \(e\left(\frac{123}{130}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{33}{130}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) |
\(\chi_{869}(249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{130}\right)\) | \(e\left(\frac{93}{130}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) |
\(\chi_{869}(270,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{11}{130}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{41}{130}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) |
\(\chi_{869}(294,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{127}{130}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) |
\(\chi_{869}(343,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{130}\right)\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{99}{130}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) |
\(\chi_{869}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{37}{130}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{67}{130}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) |
\(\chi_{869}(387,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{130}\right)\) | \(e\left(\frac{59}{130}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{19}{130}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) |
\(\chi_{869}(409,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{130}\right)\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{9}{130}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) |
\(\chi_{869}(436,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{130}\right)\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) |
\(\chi_{869}(453,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{79}{130}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) |
\(\chi_{869}(464,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{19}{130}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{59}{130}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) |
\(\chi_{869}(486,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{119}{130}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{89}{130}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) |
\(\chi_{869}(491,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) |
\(\chi_{869}(501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{130}\right)\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{21}{130}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) |
\(\chi_{869}(535,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{23}{130}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{3}{130}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) |
\(\chi_{869}(545,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{130}\right)\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{71}{130}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) |
\(\chi_{869}(567,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{121}{130}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{61}{130}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) |