from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3721, base_ring=CyclotomicField(122))
M = H._module
chi = DirichletCharacter(H, M([78]))
chi.galois_orbit()
[g,chi] = znchar(Mod(62,3721))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3721\) | |
| Conductor: | \(3721\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(61\) | 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: | $\Q(\zeta_{61})$ |
| Fixed field: | Number field defined by a degree 61 polynomial |
First 31 of 60 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3721}(62,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{61}\right)\) | \(e\left(\frac{9}{61}\right)\) | \(e\left(\frac{17}{61}\right)\) | \(e\left(\frac{31}{61}\right)\) | \(e\left(\frac{48}{61}\right)\) | \(e\left(\frac{23}{61}\right)\) | \(e\left(\frac{56}{61}\right)\) | \(e\left(\frac{18}{61}\right)\) | \(e\left(\frac{9}{61}\right)\) | \(e\left(\frac{57}{61}\right)\) |
| \(\chi_{3721}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{61}\right)\) | \(e\left(\frac{18}{61}\right)\) | \(e\left(\frac{34}{61}\right)\) | \(e\left(\frac{1}{61}\right)\) | \(e\left(\frac{35}{61}\right)\) | \(e\left(\frac{46}{61}\right)\) | \(e\left(\frac{51}{61}\right)\) | \(e\left(\frac{36}{61}\right)\) | \(e\left(\frac{18}{61}\right)\) | \(e\left(\frac{53}{61}\right)\) |
| \(\chi_{3721}(184,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{56}{61}\right)\) | \(e\left(\frac{27}{61}\right)\) | \(e\left(\frac{51}{61}\right)\) | \(e\left(\frac{32}{61}\right)\) | \(e\left(\frac{22}{61}\right)\) | \(e\left(\frac{8}{61}\right)\) | \(e\left(\frac{46}{61}\right)\) | \(e\left(\frac{54}{61}\right)\) | \(e\left(\frac{27}{61}\right)\) | \(e\left(\frac{49}{61}\right)\) |
| \(\chi_{3721}(245,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{61}\right)\) | \(e\left(\frac{36}{61}\right)\) | \(e\left(\frac{7}{61}\right)\) | \(e\left(\frac{2}{61}\right)\) | \(e\left(\frac{9}{61}\right)\) | \(e\left(\frac{31}{61}\right)\) | \(e\left(\frac{41}{61}\right)\) | \(e\left(\frac{11}{61}\right)\) | \(e\left(\frac{36}{61}\right)\) | \(e\left(\frac{45}{61}\right)\) |
| \(\chi_{3721}(306,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{61}\right)\) | \(e\left(\frac{45}{61}\right)\) | \(e\left(\frac{24}{61}\right)\) | \(e\left(\frac{33}{61}\right)\) | \(e\left(\frac{57}{61}\right)\) | \(e\left(\frac{54}{61}\right)\) | \(e\left(\frac{36}{61}\right)\) | \(e\left(\frac{29}{61}\right)\) | \(e\left(\frac{45}{61}\right)\) | \(e\left(\frac{41}{61}\right)\) |
| \(\chi_{3721}(367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{61}\right)\) | \(e\left(\frac{54}{61}\right)\) | \(e\left(\frac{41}{61}\right)\) | \(e\left(\frac{3}{61}\right)\) | \(e\left(\frac{44}{61}\right)\) | \(e\left(\frac{16}{61}\right)\) | \(e\left(\frac{31}{61}\right)\) | \(e\left(\frac{47}{61}\right)\) | \(e\left(\frac{54}{61}\right)\) | \(e\left(\frac{37}{61}\right)\) |
| \(\chi_{3721}(428,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{61}\right)\) | \(e\left(\frac{2}{61}\right)\) | \(e\left(\frac{58}{61}\right)\) | \(e\left(\frac{34}{61}\right)\) | \(e\left(\frac{31}{61}\right)\) | \(e\left(\frac{39}{61}\right)\) | \(e\left(\frac{26}{61}\right)\) | \(e\left(\frac{4}{61}\right)\) | \(e\left(\frac{2}{61}\right)\) | \(e\left(\frac{33}{61}\right)\) |
| \(\chi_{3721}(489,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{61}\right)\) | \(e\left(\frac{11}{61}\right)\) | \(e\left(\frac{14}{61}\right)\) | \(e\left(\frac{4}{61}\right)\) | \(e\left(\frac{18}{61}\right)\) | \(e\left(\frac{1}{61}\right)\) | \(e\left(\frac{21}{61}\right)\) | \(e\left(\frac{22}{61}\right)\) | \(e\left(\frac{11}{61}\right)\) | \(e\left(\frac{29}{61}\right)\) |
| \(\chi_{3721}(550,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{61}\right)\) | \(e\left(\frac{20}{61}\right)\) | \(e\left(\frac{31}{61}\right)\) | \(e\left(\frac{35}{61}\right)\) | \(e\left(\frac{5}{61}\right)\) | \(e\left(\frac{24}{61}\right)\) | \(e\left(\frac{16}{61}\right)\) | \(e\left(\frac{40}{61}\right)\) | \(e\left(\frac{20}{61}\right)\) | \(e\left(\frac{25}{61}\right)\) |
| \(\chi_{3721}(611,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{61}\right)\) | \(e\left(\frac{29}{61}\right)\) | \(e\left(\frac{48}{61}\right)\) | \(e\left(\frac{5}{61}\right)\) | \(e\left(\frac{53}{61}\right)\) | \(e\left(\frac{47}{61}\right)\) | \(e\left(\frac{11}{61}\right)\) | \(e\left(\frac{58}{61}\right)\) | \(e\left(\frac{29}{61}\right)\) | \(e\left(\frac{21}{61}\right)\) |
| \(\chi_{3721}(672,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{61}\right)\) | \(e\left(\frac{38}{61}\right)\) | \(e\left(\frac{4}{61}\right)\) | \(e\left(\frac{36}{61}\right)\) | \(e\left(\frac{40}{61}\right)\) | \(e\left(\frac{9}{61}\right)\) | \(e\left(\frac{6}{61}\right)\) | \(e\left(\frac{15}{61}\right)\) | \(e\left(\frac{38}{61}\right)\) | \(e\left(\frac{17}{61}\right)\) |
| \(\chi_{3721}(733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{61}\right)\) | \(e\left(\frac{47}{61}\right)\) | \(e\left(\frac{21}{61}\right)\) | \(e\left(\frac{6}{61}\right)\) | \(e\left(\frac{27}{61}\right)\) | \(e\left(\frac{32}{61}\right)\) | \(e\left(\frac{1}{61}\right)\) | \(e\left(\frac{33}{61}\right)\) | \(e\left(\frac{47}{61}\right)\) | \(e\left(\frac{13}{61}\right)\) |
| \(\chi_{3721}(794,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{61}\right)\) | \(e\left(\frac{56}{61}\right)\) | \(e\left(\frac{38}{61}\right)\) | \(e\left(\frac{37}{61}\right)\) | \(e\left(\frac{14}{61}\right)\) | \(e\left(\frac{55}{61}\right)\) | \(e\left(\frac{57}{61}\right)\) | \(e\left(\frac{51}{61}\right)\) | \(e\left(\frac{56}{61}\right)\) | \(e\left(\frac{9}{61}\right)\) |
| \(\chi_{3721}(855,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{61}\right)\) | \(e\left(\frac{4}{61}\right)\) | \(e\left(\frac{55}{61}\right)\) | \(e\left(\frac{7}{61}\right)\) | \(e\left(\frac{1}{61}\right)\) | \(e\left(\frac{17}{61}\right)\) | \(e\left(\frac{52}{61}\right)\) | \(e\left(\frac{8}{61}\right)\) | \(e\left(\frac{4}{61}\right)\) | \(e\left(\frac{5}{61}\right)\) |
| \(\chi_{3721}(916,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{61}\right)\) | \(e\left(\frac{13}{61}\right)\) | \(e\left(\frac{11}{61}\right)\) | \(e\left(\frac{38}{61}\right)\) | \(e\left(\frac{49}{61}\right)\) | \(e\left(\frac{40}{61}\right)\) | \(e\left(\frac{47}{61}\right)\) | \(e\left(\frac{26}{61}\right)\) | \(e\left(\frac{13}{61}\right)\) | \(e\left(\frac{1}{61}\right)\) |
| \(\chi_{3721}(977,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{61}\right)\) | \(e\left(\frac{22}{61}\right)\) | \(e\left(\frac{28}{61}\right)\) | \(e\left(\frac{8}{61}\right)\) | \(e\left(\frac{36}{61}\right)\) | \(e\left(\frac{2}{61}\right)\) | \(e\left(\frac{42}{61}\right)\) | \(e\left(\frac{44}{61}\right)\) | \(e\left(\frac{22}{61}\right)\) | \(e\left(\frac{58}{61}\right)\) |
| \(\chi_{3721}(1038,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{61}\right)\) | \(e\left(\frac{31}{61}\right)\) | \(e\left(\frac{45}{61}\right)\) | \(e\left(\frac{39}{61}\right)\) | \(e\left(\frac{23}{61}\right)\) | \(e\left(\frac{25}{61}\right)\) | \(e\left(\frac{37}{61}\right)\) | \(e\left(\frac{1}{61}\right)\) | \(e\left(\frac{31}{61}\right)\) | \(e\left(\frac{54}{61}\right)\) |
| \(\chi_{3721}(1099,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{61}\right)\) | \(e\left(\frac{40}{61}\right)\) | \(e\left(\frac{1}{61}\right)\) | \(e\left(\frac{9}{61}\right)\) | \(e\left(\frac{10}{61}\right)\) | \(e\left(\frac{48}{61}\right)\) | \(e\left(\frac{32}{61}\right)\) | \(e\left(\frac{19}{61}\right)\) | \(e\left(\frac{40}{61}\right)\) | \(e\left(\frac{50}{61}\right)\) |
| \(\chi_{3721}(1160,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{61}\right)\) | \(e\left(\frac{49}{61}\right)\) | \(e\left(\frac{18}{61}\right)\) | \(e\left(\frac{40}{61}\right)\) | \(e\left(\frac{58}{61}\right)\) | \(e\left(\frac{10}{61}\right)\) | \(e\left(\frac{27}{61}\right)\) | \(e\left(\frac{37}{61}\right)\) | \(e\left(\frac{49}{61}\right)\) | \(e\left(\frac{46}{61}\right)\) |
| \(\chi_{3721}(1221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{48}{61}\right)\) | \(e\left(\frac{58}{61}\right)\) | \(e\left(\frac{35}{61}\right)\) | \(e\left(\frac{10}{61}\right)\) | \(e\left(\frac{45}{61}\right)\) | \(e\left(\frac{33}{61}\right)\) | \(e\left(\frac{22}{61}\right)\) | \(e\left(\frac{55}{61}\right)\) | \(e\left(\frac{58}{61}\right)\) | \(e\left(\frac{42}{61}\right)\) |
| \(\chi_{3721}(1282,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{61}\right)\) | \(e\left(\frac{6}{61}\right)\) | \(e\left(\frac{52}{61}\right)\) | \(e\left(\frac{41}{61}\right)\) | \(e\left(\frac{32}{61}\right)\) | \(e\left(\frac{56}{61}\right)\) | \(e\left(\frac{17}{61}\right)\) | \(e\left(\frac{12}{61}\right)\) | \(e\left(\frac{6}{61}\right)\) | \(e\left(\frac{38}{61}\right)\) |
| \(\chi_{3721}(1343,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{61}\right)\) | \(e\left(\frac{15}{61}\right)\) | \(e\left(\frac{8}{61}\right)\) | \(e\left(\frac{11}{61}\right)\) | \(e\left(\frac{19}{61}\right)\) | \(e\left(\frac{18}{61}\right)\) | \(e\left(\frac{12}{61}\right)\) | \(e\left(\frac{30}{61}\right)\) | \(e\left(\frac{15}{61}\right)\) | \(e\left(\frac{34}{61}\right)\) |
| \(\chi_{3721}(1404,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{61}\right)\) | \(e\left(\frac{24}{61}\right)\) | \(e\left(\frac{25}{61}\right)\) | \(e\left(\frac{42}{61}\right)\) | \(e\left(\frac{6}{61}\right)\) | \(e\left(\frac{41}{61}\right)\) | \(e\left(\frac{7}{61}\right)\) | \(e\left(\frac{48}{61}\right)\) | \(e\left(\frac{24}{61}\right)\) | \(e\left(\frac{30}{61}\right)\) |
| \(\chi_{3721}(1465,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{61}\right)\) | \(e\left(\frac{33}{61}\right)\) | \(e\left(\frac{42}{61}\right)\) | \(e\left(\frac{12}{61}\right)\) | \(e\left(\frac{54}{61}\right)\) | \(e\left(\frac{3}{61}\right)\) | \(e\left(\frac{2}{61}\right)\) | \(e\left(\frac{5}{61}\right)\) | \(e\left(\frac{33}{61}\right)\) | \(e\left(\frac{26}{61}\right)\) |
| \(\chi_{3721}(1526,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{60}{61}\right)\) | \(e\left(\frac{42}{61}\right)\) | \(e\left(\frac{59}{61}\right)\) | \(e\left(\frac{43}{61}\right)\) | \(e\left(\frac{41}{61}\right)\) | \(e\left(\frac{26}{61}\right)\) | \(e\left(\frac{58}{61}\right)\) | \(e\left(\frac{23}{61}\right)\) | \(e\left(\frac{42}{61}\right)\) | \(e\left(\frac{22}{61}\right)\) |
| \(\chi_{3721}(1587,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{61}\right)\) | \(e\left(\frac{51}{61}\right)\) | \(e\left(\frac{15}{61}\right)\) | \(e\left(\frac{13}{61}\right)\) | \(e\left(\frac{28}{61}\right)\) | \(e\left(\frac{49}{61}\right)\) | \(e\left(\frac{53}{61}\right)\) | \(e\left(\frac{41}{61}\right)\) | \(e\left(\frac{51}{61}\right)\) | \(e\left(\frac{18}{61}\right)\) |
| \(\chi_{3721}(1648,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{61}\right)\) | \(e\left(\frac{60}{61}\right)\) | \(e\left(\frac{32}{61}\right)\) | \(e\left(\frac{44}{61}\right)\) | \(e\left(\frac{15}{61}\right)\) | \(e\left(\frac{11}{61}\right)\) | \(e\left(\frac{48}{61}\right)\) | \(e\left(\frac{59}{61}\right)\) | \(e\left(\frac{60}{61}\right)\) | \(e\left(\frac{14}{61}\right)\) |
| \(\chi_{3721}(1709,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{61}\right)\) | \(e\left(\frac{8}{61}\right)\) | \(e\left(\frac{49}{61}\right)\) | \(e\left(\frac{14}{61}\right)\) | \(e\left(\frac{2}{61}\right)\) | \(e\left(\frac{34}{61}\right)\) | \(e\left(\frac{43}{61}\right)\) | \(e\left(\frac{16}{61}\right)\) | \(e\left(\frac{8}{61}\right)\) | \(e\left(\frac{10}{61}\right)\) |
| \(\chi_{3721}(1770,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{61}\right)\) | \(e\left(\frac{17}{61}\right)\) | \(e\left(\frac{5}{61}\right)\) | \(e\left(\frac{45}{61}\right)\) | \(e\left(\frac{50}{61}\right)\) | \(e\left(\frac{57}{61}\right)\) | \(e\left(\frac{38}{61}\right)\) | \(e\left(\frac{34}{61}\right)\) | \(e\left(\frac{17}{61}\right)\) | \(e\left(\frac{6}{61}\right)\) |
| \(\chi_{3721}(1831,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{61}\right)\) | \(e\left(\frac{26}{61}\right)\) | \(e\left(\frac{22}{61}\right)\) | \(e\left(\frac{15}{61}\right)\) | \(e\left(\frac{37}{61}\right)\) | \(e\left(\frac{19}{61}\right)\) | \(e\left(\frac{33}{61}\right)\) | \(e\left(\frac{52}{61}\right)\) | \(e\left(\frac{26}{61}\right)\) | \(e\left(\frac{2}{61}\right)\) |
| \(\chi_{3721}(1892,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{61}\right)\) | \(e\left(\frac{35}{61}\right)\) | \(e\left(\frac{39}{61}\right)\) | \(e\left(\frac{46}{61}\right)\) | \(e\left(\frac{24}{61}\right)\) | \(e\left(\frac{42}{61}\right)\) | \(e\left(\frac{28}{61}\right)\) | \(e\left(\frac{9}{61}\right)\) | \(e\left(\frac{35}{61}\right)\) | \(e\left(\frac{59}{61}\right)\) |