from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2230, base_ring=CyclotomicField(74))
M = H._module
chi = DirichletCharacter(H, M([37,66]))
chi.galois_orbit()
[g,chi] = znchar(Mod(49,2230))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2230\) | |
| Conductor: | \(1115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(74\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1115.p | 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: | $\Q(\zeta_{37})$ |
| Fixed field: | Number field defined by a degree 74 polynomial |
First 31 of 36 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2230}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{74}\right)\) | \(e\left(\frac{59}{74}\right)\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{16}{37}\right)\) | \(e\left(\frac{45}{74}\right)\) | \(e\left(\frac{69}{74}\right)\) | \(e\left(\frac{15}{37}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{65}{74}\right)\) | \(e\left(\frac{13}{74}\right)\) |
| \(\chi_{2230}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{74}\right)\) | \(e\left(\frac{27}{74}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{23}{37}\right)\) | \(e\left(\frac{67}{74}\right)\) | \(e\left(\frac{9}{74}\right)\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{31}{74}\right)\) | \(e\left(\frac{21}{74}\right)\) |
| \(\chi_{2230}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{74}\right)\) | \(e\left(\frac{45}{74}\right)\) | \(e\left(\frac{24}{37}\right)\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{13}{74}\right)\) | \(e\left(\frac{15}{74}\right)\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{16}{37}\right)\) | \(e\left(\frac{27}{74}\right)\) | \(e\left(\frac{35}{74}\right)\) |
| \(\chi_{2230}(239,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{74}\right)\) | \(e\left(\frac{43}{74}\right)\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{19}{74}\right)\) | \(e\left(\frac{39}{74}\right)\) | \(e\left(\frac{31}{37}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{11}{74}\right)\) | \(e\left(\frac{17}{74}\right)\) |
| \(\chi_{2230}(279,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{74}\right)\) | \(e\left(\frac{71}{74}\right)\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{9}{74}\right)\) | \(e\left(\frac{73}{74}\right)\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{31}{37}\right)\) | \(e\left(\frac{13}{74}\right)\) | \(e\left(\frac{47}{74}\right)\) |
| \(\chi_{2230}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{74}\right)\) | \(e\left(\frac{69}{74}\right)\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{15}{74}\right)\) | \(e\left(\frac{23}{74}\right)\) | \(e\left(\frac{5}{37}\right)\) | \(e\left(\frac{27}{37}\right)\) | \(e\left(\frac{71}{74}\right)\) | \(e\left(\frac{29}{74}\right)\) |
| \(\chi_{2230}(359,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{74}\right)\) | \(e\left(\frac{39}{74}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{25}{37}\right)\) | \(e\left(\frac{31}{74}\right)\) | \(e\left(\frac{13}{74}\right)\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{53}{74}\right)\) | \(e\left(\frac{55}{74}\right)\) |
| \(\chi_{2230}(419,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{74}\right)\) | \(e\left(\frac{25}{74}\right)\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{73}{74}\right)\) | \(e\left(\frac{33}{74}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{15}{74}\right)\) | \(e\left(\frac{3}{74}\right)\) |
| \(\chi_{2230}(479,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{74}\right)\) | \(e\left(\frac{49}{74}\right)\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{1}{74}\right)\) | \(e\left(\frac{41}{74}\right)\) | \(e\left(\frac{25}{37}\right)\) | \(e\left(\frac{24}{37}\right)\) | \(e\left(\frac{59}{74}\right)\) | \(e\left(\frac{71}{74}\right)\) |
| \(\chi_{2230}(699,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{74}\right)\) | \(e\left(\frac{67}{74}\right)\) | \(e\left(\frac{16}{37}\right)\) | \(e\left(\frac{5}{37}\right)\) | \(e\left(\frac{21}{74}\right)\) | \(e\left(\frac{47}{74}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{23}{37}\right)\) | \(e\left(\frac{55}{74}\right)\) | \(e\left(\frac{11}{74}\right)\) |
| \(\chi_{2230}(729,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{74}\right)\) | \(e\left(\frac{13}{74}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{33}{37}\right)\) | \(e\left(\frac{35}{74}\right)\) | \(e\left(\frac{29}{74}\right)\) | \(e\left(\frac{24}{37}\right)\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{67}{74}\right)\) | \(e\left(\frac{43}{74}\right)\) |
| \(\chi_{2230}(789,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{74}\right)\) | \(e\left(\frac{33}{74}\right)\) | \(e\left(\frac{25}{37}\right)\) | \(e\left(\frac{24}{37}\right)\) | \(e\left(\frac{49}{74}\right)\) | \(e\left(\frac{11}{74}\right)\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{5}{74}\right)\) | \(e\left(\frac{1}{74}\right)\) |
| \(\chi_{2230}(879,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{74}\right)\) | \(e\left(\frac{41}{74}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{25}{74}\right)\) | \(e\left(\frac{63}{74}\right)\) | \(e\left(\frac{33}{37}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{69}{74}\right)\) | \(e\left(\frac{73}{74}\right)\) |
| \(\chi_{2230}(899,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{74}\right)\) | \(e\left(\frac{11}{74}\right)\) | \(e\left(\frac{33}{37}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{41}{74}\right)\) | \(e\left(\frac{53}{74}\right)\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{51}{74}\right)\) | \(e\left(\frac{25}{74}\right)\) |
| \(\chi_{2230}(909,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{74}\right)\) | \(e\left(\frac{53}{74}\right)\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{15}{37}\right)\) | \(e\left(\frac{63}{74}\right)\) | \(e\left(\frac{67}{74}\right)\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{32}{37}\right)\) | \(e\left(\frac{17}{74}\right)\) | \(e\left(\frac{33}{74}\right)\) |
| \(\chi_{2230}(1089,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{74}\right)\) | \(e\left(\frac{61}{74}\right)\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{39}{74}\right)\) | \(e\left(\frac{45}{74}\right)\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{7}{74}\right)\) | \(e\left(\frac{31}{74}\right)\) |
| \(\chi_{2230}(1119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{74}\right)\) | \(e\left(\frac{3}{74}\right)\) | \(e\left(\frac{9}{37}\right)\) | \(e\left(\frac{19}{37}\right)\) | \(e\left(\frac{65}{74}\right)\) | \(e\left(\frac{1}{74}\right)\) | \(e\left(\frac{34}{37}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{61}{74}\right)\) | \(e\left(\frac{27}{74}\right)\) |
| \(\chi_{2230}(1129,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{74}\right)\) | \(e\left(\frac{31}{74}\right)\) | \(e\left(\frac{19}{37}\right)\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{55}{74}\right)\) | \(e\left(\frac{35}{74}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{25}{37}\right)\) | \(e\left(\frac{63}{74}\right)\) | \(e\left(\frac{57}{74}\right)\) |
| \(\chi_{2230}(1149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{74}\right)\) | \(e\left(\frac{73}{74}\right)\) | \(e\left(\frac{34}{37}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{3}{74}\right)\) | \(e\left(\frac{49}{74}\right)\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{29}{74}\right)\) | \(e\left(\frac{65}{74}\right)\) |
| \(\chi_{2230}(1179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{74}\right)\) | \(e\left(\frac{9}{74}\right)\) | \(e\left(\frac{27}{37}\right)\) | \(e\left(\frac{20}{37}\right)\) | \(e\left(\frac{47}{74}\right)\) | \(e\left(\frac{3}{74}\right)\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{35}{74}\right)\) | \(e\left(\frac{7}{74}\right)\) |
| \(\chi_{2230}(1279,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{74}\right)\) | \(e\left(\frac{1}{74}\right)\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{31}{37}\right)\) | \(e\left(\frac{71}{74}\right)\) | \(e\left(\frac{25}{74}\right)\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{45}{74}\right)\) | \(e\left(\frac{9}{74}\right)\) |
| \(\chi_{2230}(1379,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{74}\right)\) | \(e\left(\frac{35}{74}\right)\) | \(e\left(\frac{31}{37}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{43}{74}\right)\) | \(e\left(\frac{61}{74}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{33}{37}\right)\) | \(e\left(\frac{21}{74}\right)\) | \(e\left(\frac{19}{74}\right)\) |
| \(\chi_{2230}(1509,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{74}\right)\) | \(e\left(\frac{7}{74}\right)\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{32}{37}\right)\) | \(e\left(\frac{53}{74}\right)\) | \(e\left(\frac{27}{74}\right)\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{19}{74}\right)\) | \(e\left(\frac{63}{74}\right)\) |
| \(\chi_{2230}(1569,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{74}\right)\) | \(e\left(\frac{23}{74}\right)\) | \(e\left(\frac{32}{37}\right)\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{5}{74}\right)\) | \(e\left(\frac{57}{74}\right)\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{9}{37}\right)\) | \(e\left(\frac{73}{74}\right)\) | \(e\left(\frac{59}{74}\right)\) |
| \(\chi_{2230}(1589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{74}\right)\) | \(e\left(\frac{51}{74}\right)\) | \(e\left(\frac{5}{37}\right)\) | \(e\left(\frac{27}{37}\right)\) | \(e\left(\frac{69}{74}\right)\) | \(e\left(\frac{17}{74}\right)\) | \(e\left(\frac{23}{37}\right)\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{1}{74}\right)\) | \(e\left(\frac{15}{74}\right)\) |
| \(\chi_{2230}(1629,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{74}\right)\) | \(e\left(\frac{19}{74}\right)\) | \(e\left(\frac{20}{37}\right)\) | \(e\left(\frac{34}{37}\right)\) | \(e\left(\frac{17}{74}\right)\) | \(e\left(\frac{31}{74}\right)\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{41}{74}\right)\) | \(e\left(\frac{23}{74}\right)\) |
| \(\chi_{2230}(1659,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{74}\right)\) | \(e\left(\frac{5}{74}\right)\) | \(e\left(\frac{15}{37}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{59}{74}\right)\) | \(e\left(\frac{51}{74}\right)\) | \(e\left(\frac{32}{37}\right)\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{3}{74}\right)\) | \(e\left(\frac{45}{74}\right)\) |
| \(\chi_{2230}(1689,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{74}\right)\) | \(e\left(\frac{29}{74}\right)\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{61}{74}\right)\) | \(e\left(\frac{59}{74}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{47}{74}\right)\) | \(e\left(\frac{39}{74}\right)\) |
| \(\chi_{2230}(1799,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{74}\right)\) | \(e\left(\frac{47}{74}\right)\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{7}{74}\right)\) | \(e\left(\frac{65}{74}\right)\) | \(e\left(\frac{27}{37}\right)\) | \(e\left(\frac{20}{37}\right)\) | \(e\left(\frac{43}{74}\right)\) | \(e\left(\frac{53}{74}\right)\) |
| \(\chi_{2230}(1889,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{74}\right)\) | \(e\left(\frac{21}{74}\right)\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{11}{74}\right)\) | \(e\left(\frac{7}{74}\right)\) | \(e\left(\frac{16}{37}\right)\) | \(e\left(\frac{5}{37}\right)\) | \(e\left(\frac{57}{74}\right)\) | \(e\left(\frac{41}{74}\right)\) |
| \(\chi_{2230}(1899,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{74}\right)\) | \(e\left(\frac{65}{74}\right)\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{27}{74}\right)\) | \(e\left(\frac{71}{74}\right)\) | \(e\left(\frac{9}{37}\right)\) | \(e\left(\frac{19}{37}\right)\) | \(e\left(\frac{39}{74}\right)\) | \(e\left(\frac{67}{74}\right)\) |