from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8304, base_ring=CyclotomicField(172))
M = H._module
chi = DirichletCharacter(H, M([86,129,0,33]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,8304))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8304\) | |
| Conductor: | \(2768\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(172\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2768.bc | 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: | $\Q(\zeta_{172})$ |
| Fixed field: | Number field defined by a degree 172 polynomial (not computed) |
First 31 of 84 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8304}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{39}{172}\right)\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{33}{172}\right)\) | \(e\left(\frac{1}{172}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{151}{172}\right)\) | \(e\left(\frac{3}{43}\right)\) |
| \(\chi_{8304}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{153}{172}\right)\) | \(e\left(\frac{45}{86}\right)\) | \(e\left(\frac{103}{172}\right)\) | \(e\left(\frac{123}{172}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{169}{172}\right)\) | \(e\left(\frac{25}{43}\right)\) |
| \(\chi_{8304}(259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{163}{172}\right)\) | \(e\left(\frac{53}{86}\right)\) | \(e\left(\frac{85}{172}\right)\) | \(e\left(\frac{13}{172}\right)\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{71}{172}\right)\) | \(e\left(\frac{39}{43}\right)\) |
| \(\chi_{8304}(283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{137}{172}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{63}{172}\right)\) | \(e\left(\frac{127}{172}\right)\) | \(e\left(\frac{29}{86}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{85}{172}\right)\) | \(e\left(\frac{37}{43}\right)\) |
| \(\chi_{8304}(451,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{159}{172}\right)\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{161}{172}\right)\) | \(e\left(\frac{57}{172}\right)\) | \(e\left(\frac{55}{86}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{7}{172}\right)\) | \(e\left(\frac{42}{43}\right)\) |
| \(\chi_{8304}(475,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{53}{172}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{111}{172}\right)\) | \(e\left(\frac{19}{172}\right)\) | \(e\left(\frac{47}{86}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{117}{172}\right)\) | \(e\left(\frac{14}{43}\right)\) |
| \(\chi_{8304}(547,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{99}{172}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{97}{172}\right)\) | \(e\left(\frac{29}{172}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{79}{172}\right)\) | \(e\left(\frac{1}{43}\right)\) |
| \(\chi_{8304}(763,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{49}{172}\right)\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{15}{172}\right)\) | \(e\left(\frac{63}{172}\right)\) | \(e\left(\frac{11}{86}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{53}{172}\right)\) | \(e\left(\frac{17}{43}\right)\) |
| \(\chi_{8304}(835,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{87}{172}\right)\) | \(e\left(\frac{61}{86}\right)\) | \(e\left(\frac{153}{172}\right)\) | \(e\left(\frac{161}{172}\right)\) | \(e\left(\frac{9}{86}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{59}{172}\right)\) | \(e\left(\frac{10}{43}\right)\) |
| \(\chi_{8304}(907,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{161}{172}\right)\) | \(e\left(\frac{17}{86}\right)\) | \(e\left(\frac{123}{172}\right)\) | \(e\left(\frac{35}{172}\right)\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{125}{172}\right)\) | \(e\left(\frac{19}{43}\right)\) |
| \(\chi_{8304}(979,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{119}{172}\right)\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{61}{172}\right)\) | \(e\left(\frac{153}{172}\right)\) | \(e\left(\frac{39}{86}\right)\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{55}{172}\right)\) | \(e\left(\frac{29}{43}\right)\) |
| \(\chi_{8304}(1027,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{35}{172}\right)\) | \(e\left(\frac{71}{86}\right)\) | \(e\left(\frac{109}{172}\right)\) | \(e\left(\frac{45}{172}\right)\) | \(e\left(\frac{57}{86}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{87}{172}\right)\) | \(e\left(\frac{6}{43}\right)\) |
| \(\chi_{8304}(1315,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{79}{172}\right)\) | \(e\left(\frac{3}{86}\right)\) | \(e\left(\frac{133}{172}\right)\) | \(e\left(\frac{77}{172}\right)\) | \(e\left(\frac{23}{86}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{103}{172}\right)\) | \(e\left(\frac{16}{43}\right)\) |
| \(\chi_{8304}(1339,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{149}{172}\right)\) | \(e\left(\frac{59}{86}\right)\) | \(e\left(\frac{7}{172}\right)\) | \(e\left(\frac{167}{172}\right)\) | \(e\left(\frac{51}{86}\right)\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{105}{172}\right)\) | \(e\left(\frac{28}{43}\right)\) |
| \(\chi_{8304}(1387,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{157}{172}\right)\) | \(e\left(\frac{31}{86}\right)\) | \(e\left(\frac{27}{172}\right)\) | \(e\left(\frac{79}{172}\right)\) | \(e\left(\frac{37}{86}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{61}{172}\right)\) | \(e\left(\frac{22}{43}\right)\) |
| \(\chi_{8304}(1411,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{127}{172}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{81}{172}\right)\) | \(e\left(\frac{65}{172}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{11}{172}\right)\) | \(e\left(\frac{23}{43}\right)\) |
| \(\chi_{8304}(1459,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{171}{172}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{105}{172}\right)\) | \(e\left(\frac{97}{172}\right)\) | \(e\left(\frac{77}{86}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{27}{172}\right)\) | \(e\left(\frac{33}{43}\right)\) |
| \(\chi_{8304}(1507,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{23}{172}\right)\) | \(e\left(\frac{27}{86}\right)\) | \(e\left(\frac{165}{172}\right)\) | \(e\left(\frac{5}{172}\right)\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{67}{172}\right)\) | \(e\left(\frac{15}{43}\right)\) |
| \(\chi_{8304}(1603,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{103}{172}\right)\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{21}{172}\right)\) | \(e\left(\frac{157}{172}\right)\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{143}{172}\right)\) | \(e\left(\frac{41}{43}\right)\) |
| \(\chi_{8304}(1627,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{97}{172}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{135}{172}\right)\) | \(e\left(\frac{51}{172}\right)\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{133}{172}\right)\) | \(e\left(\frac{24}{43}\right)\) |
| \(\chi_{8304}(1651,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{155}{172}\right)\) | \(e\left(\frac{81}{86}\right)\) | \(e\left(\frac{65}{172}\right)\) | \(e\left(\frac{101}{172}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{115}{172}\right)\) | \(e\left(\frac{2}{43}\right)\) |
| \(\chi_{8304}(1747,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{55}{172}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{73}{172}\right)\) | \(e\left(\frac{169}{172}\right)\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{63}{172}\right)\) | \(e\left(\frac{34}{43}\right)\) |
| \(\chi_{8304}(1795,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{59}{172}\right)\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{169}{172}\right)\) | \(e\left(\frac{125}{172}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{127}{172}\right)\) | \(e\left(\frac{31}{43}\right)\) |
| \(\chi_{8304}(2011,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{145}{172}\right)\) | \(e\left(\frac{73}{86}\right)\) | \(e\left(\frac{83}{172}\right)\) | \(e\left(\frac{39}{172}\right)\) | \(e\left(\frac{15}{86}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{41}{172}\right)\) | \(e\left(\frac{31}{43}\right)\) |
| \(\chi_{8304}(2059,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{141}{172}\right)\) | \(e\left(\frac{1}{86}\right)\) | \(e\left(\frac{159}{172}\right)\) | \(e\left(\frac{83}{172}\right)\) | \(e\left(\frac{65}{86}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{149}{172}\right)\) | \(e\left(\frac{34}{43}\right)\) |
| \(\chi_{8304}(2155,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{69}{172}\right)\) | \(e\left(\frac{81}{86}\right)\) | \(e\left(\frac{151}{172}\right)\) | \(e\left(\frac{15}{172}\right)\) | \(e\left(\frac{19}{86}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{29}{172}\right)\) | \(e\left(\frac{2}{43}\right)\) |
| \(\chi_{8304}(2179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{11}{172}\right)\) | \(e\left(\frac{69}{86}\right)\) | \(e\left(\frac{49}{172}\right)\) | \(e\left(\frac{137}{172}\right)\) | \(e\left(\frac{13}{86}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{47}{172}\right)\) | \(e\left(\frac{24}{43}\right)\) |
| \(\chi_{8304}(2203,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{17}{172}\right)\) | \(e\left(\frac{5}{86}\right)\) | \(e\left(\frac{107}{172}\right)\) | \(e\left(\frac{71}{172}\right)\) | \(e\left(\frac{67}{86}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{57}{172}\right)\) | \(e\left(\frac{41}{43}\right)\) |
| \(\chi_{8304}(2299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{109}{172}\right)\) | \(e\left(\frac{27}{86}\right)\) | \(e\left(\frac{79}{172}\right)\) | \(e\left(\frac{91}{172}\right)\) | \(e\left(\frac{35}{86}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{153}{172}\right)\) | \(e\left(\frac{15}{43}\right)\) |
| \(\chi_{8304}(2347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{85}{172}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{19}{172}\right)\) | \(e\left(\frac{11}{172}\right)\) | \(e\left(\frac{77}{86}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{113}{172}\right)\) | \(e\left(\frac{33}{43}\right)\) |
| \(\chi_{8304}(2395,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{41}{172}\right)\) | \(e\left(\frac{7}{86}\right)\) | \(e\left(\frac{167}{172}\right)\) | \(e\left(\frac{151}{172}\right)\) | \(e\left(\frac{25}{86}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{97}{172}\right)\) | \(e\left(\frac{23}{43}\right)\) |