from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([78,5]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1859))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(1859\) | |
Conductor: | \(1859\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(780\) | 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: | $\Q(\zeta_{780})$ |
Fixed field: | Number field defined by a degree 780 polynomial (not computed) |
First 31 of 192 characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{1859}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{780}\right)\) | \(e\left(\frac{116}{195}\right)\) | \(e\left(\frac{83}{390}\right)\) | \(e\left(\frac{119}{260}\right)\) | \(e\left(\frac{547}{780}\right)\) | \(e\left(\frac{301}{780}\right)\) | \(e\left(\frac{83}{260}\right)\) | \(e\left(\frac{37}{195}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{21}{26}\right)\) |
\(\chi_{1859}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{547}{780}\right)\) | \(e\left(\frac{109}{195}\right)\) | \(e\left(\frac{157}{390}\right)\) | \(e\left(\frac{211}{260}\right)\) | \(e\left(\frac{203}{780}\right)\) | \(e\left(\frac{29}{780}\right)\) | \(e\left(\frac{27}{260}\right)\) | \(e\left(\frac{23}{195}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{25}{26}\right)\) |
\(\chi_{1859}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{301}{780}\right)\) | \(e\left(\frac{127}{195}\right)\) | \(e\left(\frac{301}{390}\right)\) | \(e\left(\frac{253}{260}\right)\) | \(e\left(\frac{29}{780}\right)\) | \(e\left(\frac{227}{780}\right)\) | \(e\left(\frac{41}{260}\right)\) | \(e\left(\frac{59}{195}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{11}{26}\right)\) |
\(\chi_{1859}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{713}{780}\right)\) | \(e\left(\frac{146}{195}\right)\) | \(e\left(\frac{323}{390}\right)\) | \(e\left(\frac{189}{260}\right)\) | \(e\left(\frac{517}{780}\right)\) | \(e\left(\frac{631}{780}\right)\) | \(e\left(\frac{193}{260}\right)\) | \(e\left(\frac{97}{195}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{15}{26}\right)\) |
\(\chi_{1859}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{467}{780}\right)\) | \(e\left(\frac{164}{195}\right)\) | \(e\left(\frac{77}{390}\right)\) | \(e\left(\frac{231}{260}\right)\) | \(e\left(\frac{343}{780}\right)\) | \(e\left(\frac{49}{780}\right)\) | \(e\left(\frac{207}{260}\right)\) | \(e\left(\frac{133}{195}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{1}{26}\right)\) |
\(\chi_{1859}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{659}{780}\right)\) | \(e\left(\frac{188}{195}\right)\) | \(e\left(\frac{269}{390}\right)\) | \(e\left(\frac{27}{260}\right)\) | \(e\left(\frac{631}{780}\right)\) | \(e\left(\frac{313}{780}\right)\) | \(e\left(\frac{139}{260}\right)\) | \(e\left(\frac{181}{195}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{17}{26}\right)\) |
\(\chi_{1859}(46,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{733}{780}\right)\) | \(e\left(\frac{181}{195}\right)\) | \(e\left(\frac{343}{390}\right)\) | \(e\left(\frac{249}{260}\right)\) | \(e\left(\frac{677}{780}\right)\) | \(e\left(\frac{431}{780}\right)\) | \(e\left(\frac{213}{260}\right)\) | \(e\left(\frac{167}{195}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{21}{26}\right)\) |
\(\chi_{1859}(50,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{780}\right)\) | \(e\left(\frac{59}{195}\right)\) | \(e\left(\frac{17}{390}\right)\) | \(e\left(\frac{181}{260}\right)\) | \(e\left(\frac{253}{780}\right)\) | \(e\left(\frac{259}{780}\right)\) | \(e\left(\frac{17}{260}\right)\) | \(e\left(\frac{118}{195}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{9}{26}\right)\) |
\(\chi_{1859}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{449}{780}\right)\) | \(e\left(\frac{113}{195}\right)\) | \(e\left(\frac{59}{390}\right)\) | \(e\left(\frac{177}{260}\right)\) | \(e\left(\frac{121}{780}\right)\) | \(e\left(\frac{463}{780}\right)\) | \(e\left(\frac{189}{260}\right)\) | \(e\left(\frac{31}{195}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{19}{26}\right)\) |
\(\chi_{1859}(72,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{397}{780}\right)\) | \(e\left(\frac{139}{195}\right)\) | \(e\left(\frac{7}{390}\right)\) | \(e\left(\frac{21}{260}\right)\) | \(e\left(\frac{173}{780}\right)\) | \(e\left(\frac{359}{780}\right)\) | \(e\left(\frac{137}{260}\right)\) | \(e\left(\frac{83}{195}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{19}{26}\right)\) |
\(\chi_{1859}(84,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{780}\right)\) | \(e\left(\frac{157}{195}\right)\) | \(e\left(\frac{151}{390}\right)\) | \(e\left(\frac{63}{260}\right)\) | \(e\left(\frac{779}{780}\right)\) | \(e\left(\frac{557}{780}\right)\) | \(e\left(\frac{151}{260}\right)\) | \(e\left(\frac{119}{195}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{5}{26}\right)\) |
\(\chi_{1859}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{780}\right)\) | \(e\left(\frac{118}{195}\right)\) | \(e\left(\frac{229}{390}\right)\) | \(e\left(\frac{37}{260}\right)\) | \(e\left(\frac{701}{780}\right)\) | \(e\left(\frac{323}{780}\right)\) | \(e\left(\frac{229}{260}\right)\) | \(e\left(\frac{41}{195}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{5}{26}\right)\) |
\(\chi_{1859}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{371}{780}\right)\) | \(e\left(\frac{152}{195}\right)\) | \(e\left(\frac{371}{390}\right)\) | \(e\left(\frac{203}{260}\right)\) | \(e\left(\frac{199}{780}\right)\) | \(e\left(\frac{697}{780}\right)\) | \(e\left(\frac{111}{260}\right)\) | \(e\left(\frac{109}{195}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{19}{26}\right)\) |
\(\chi_{1859}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{343}{780}\right)\) | \(e\left(\frac{181}{195}\right)\) | \(e\left(\frac{343}{390}\right)\) | \(e\left(\frac{119}{260}\right)\) | \(e\left(\frac{287}{780}\right)\) | \(e\left(\frac{41}{780}\right)\) | \(e\left(\frac{83}{260}\right)\) | \(e\left(\frac{167}{195}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{21}{26}\right)\) |
\(\chi_{1859}(128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{581}{780}\right)\) | \(e\left(\frac{32}{195}\right)\) | \(e\left(\frac{191}{390}\right)\) | \(e\left(\frac{53}{260}\right)\) | \(e\left(\frac{709}{780}\right)\) | \(e\left(\frac{547}{780}\right)\) | \(e\left(\frac{61}{260}\right)\) | \(e\left(\frac{64}{195}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{17}{26}\right)\) |
\(\chi_{1859}(145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{323}{780}\right)\) | \(e\left(\frac{146}{195}\right)\) | \(e\left(\frac{323}{390}\right)\) | \(e\left(\frac{59}{260}\right)\) | \(e\left(\frac{127}{780}\right)\) | \(e\left(\frac{241}{780}\right)\) | \(e\left(\frac{63}{260}\right)\) | \(e\left(\frac{97}{195}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{15}{26}\right)\) |
\(\chi_{1859}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{367}{780}\right)\) | \(e\left(\frac{184}{195}\right)\) | \(e\left(\frac{367}{390}\right)\) | \(e\left(\frac{191}{260}\right)\) | \(e\left(\frac{323}{780}\right)\) | \(e\left(\frac{269}{780}\right)\) | \(e\left(\frac{107}{260}\right)\) | \(e\left(\frac{173}{195}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{23}{26}\right)\) |
\(\chi_{1859}(162,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{379}{780}\right)\) | \(e\left(\frac{88}{195}\right)\) | \(e\left(\frac{379}{390}\right)\) | \(e\left(\frac{227}{260}\right)\) | \(e\left(\frac{731}{780}\right)\) | \(e\left(\frac{773}{780}\right)\) | \(e\left(\frac{119}{260}\right)\) | \(e\left(\frac{176}{195}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{11}{26}\right)\) |
\(\chi_{1859}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{473}{780}\right)\) | \(e\left(\frac{116}{195}\right)\) | \(e\left(\frac{83}{390}\right)\) | \(e\left(\frac{249}{260}\right)\) | \(e\left(\frac{157}{780}\right)\) | \(e\left(\frac{691}{780}\right)\) | \(e\left(\frac{213}{260}\right)\) | \(e\left(\frac{37}{195}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{21}{26}\right)\) |
\(\chi_{1859}(171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{707}{780}\right)\) | \(e\left(\frac{194}{195}\right)\) | \(e\left(\frac{317}{390}\right)\) | \(e\left(\frac{171}{260}\right)\) | \(e\left(\frac{703}{780}\right)\) | \(e\left(\frac{769}{780}\right)\) | \(e\left(\frac{187}{260}\right)\) | \(e\left(\frac{193}{195}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{21}{26}\right)\) |
\(\chi_{1859}(184,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{780}\right)\) | \(e\left(\frac{23}{195}\right)\) | \(e\left(\frac{119}{390}\right)\) | \(e\left(\frac{227}{260}\right)\) | \(e\left(\frac{211}{780}\right)\) | \(e\left(\frac{253}{780}\right)\) | \(e\left(\frac{119}{260}\right)\) | \(e\left(\frac{46}{195}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{11}{26}\right)\) |
\(\chi_{1859}(189,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{780}\right)\) | \(e\left(\frac{106}{195}\right)\) | \(e\left(\frac{133}{390}\right)\) | \(e\left(\frac{9}{260}\right)\) | \(e\left(\frac{557}{780}\right)\) | \(e\left(\frac{191}{780}\right)\) | \(e\left(\frac{133}{260}\right)\) | \(e\left(\frac{17}{195}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{23}{26}\right)\) |
\(\chi_{1859}(193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{557}{780}\right)\) | \(e\left(\frac{29}{195}\right)\) | \(e\left(\frac{167}{390}\right)\) | \(e\left(\frac{241}{260}\right)\) | \(e\left(\frac{673}{780}\right)\) | \(e\left(\frac{319}{780}\right)\) | \(e\left(\frac{37}{260}\right)\) | \(e\left(\frac{58}{195}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{15}{26}\right)\) |
\(\chi_{1859}(206,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{780}\right)\) | \(e\left(\frac{83}{195}\right)\) | \(e\left(\frac{209}{390}\right)\) | \(e\left(\frac{237}{260}\right)\) | \(e\left(\frac{541}{780}\right)\) | \(e\left(\frac{523}{780}\right)\) | \(e\left(\frac{209}{260}\right)\) | \(e\left(\frac{166}{195}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{25}{26}\right)\) |
\(\chi_{1859}(215,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{577}{780}\right)\) | \(e\left(\frac{64}{195}\right)\) | \(e\left(\frac{187}{390}\right)\) | \(e\left(\frac{41}{260}\right)\) | \(e\left(\frac{53}{780}\right)\) | \(e\left(\frac{119}{780}\right)\) | \(e\left(\frac{57}{260}\right)\) | \(e\left(\frac{128}{195}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{21}{26}\right)\) |
\(\chi_{1859}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{751}{780}\right)\) | \(e\left(\frac{37}{195}\right)\) | \(e\left(\frac{361}{390}\right)\) | \(e\left(\frac{43}{260}\right)\) | \(e\left(\frac{119}{780}\right)\) | \(e\left(\frac{17}{780}\right)\) | \(e\left(\frac{231}{260}\right)\) | \(e\left(\frac{74}{195}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{3}{26}\right)\) |
\(\chi_{1859}(228,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{409}{780}\right)\) | \(e\left(\frac{43}{195}\right)\) | \(e\left(\frac{19}{390}\right)\) | \(e\left(\frac{57}{260}\right)\) | \(e\left(\frac{581}{780}\right)\) | \(e\left(\frac{83}{780}\right)\) | \(e\left(\frac{149}{260}\right)\) | \(e\left(\frac{86}{195}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{7}{26}\right)\) |
\(\chi_{1859}(266,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{780}\right)\) | \(e\left(\frac{61}{195}\right)\) | \(e\left(\frac{163}{390}\right)\) | \(e\left(\frac{99}{260}\right)\) | \(e\left(\frac{407}{780}\right)\) | \(e\left(\frac{281}{780}\right)\) | \(e\left(\frac{163}{260}\right)\) | \(e\left(\frac{122}{195}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{19}{26}\right)\) |
\(\chi_{1859}(271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{341}{780}\right)\) | \(e\left(\frac{2}{195}\right)\) | \(e\left(\frac{341}{390}\right)\) | \(e\left(\frac{113}{260}\right)\) | \(e\left(\frac{349}{780}\right)\) | \(e\left(\frac{607}{780}\right)\) | \(e\left(\frac{81}{260}\right)\) | \(e\left(\frac{4}{195}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{23}{26}\right)\) |
\(\chi_{1859}(288,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{563}{780}\right)\) | \(e\left(\frac{176}{195}\right)\) | \(e\left(\frac{173}{390}\right)\) | \(e\left(\frac{259}{260}\right)\) | \(e\left(\frac{487}{780}\right)\) | \(e\left(\frac{181}{780}\right)\) | \(e\left(\frac{43}{260}\right)\) | \(e\left(\frac{157}{195}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{9}{26}\right)\) |
\(\chi_{1859}(292,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{187}{780}\right)\) | \(e\left(\frac{64}{195}\right)\) | \(e\left(\frac{187}{390}\right)\) | \(e\left(\frac{171}{260}\right)\) | \(e\left(\frac{443}{780}\right)\) | \(e\left(\frac{509}{780}\right)\) | \(e\left(\frac{187}{260}\right)\) | \(e\left(\frac{128}{195}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{21}{26}\right)\) |