from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(739, base_ring=CyclotomicField(82))
M = H._module
chi = DirichletCharacter(H, M([6]))
chi.galois_orbit()
[g,chi] = znchar(Mod(20,739))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(739\) | |
| Conductor: | \(739\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(41\) | 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: | $\Q(\zeta_{41})$ |
| Fixed field: | Number field defined by a degree 41 polynomial |
First 31 of 40 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{739}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{24}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{22}{41}\right)\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{15}{41}\right)\) |
| \(\chi_{739}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{41}\right)\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{40}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{20}{41}\right)\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{23}{41}\right)\) | \(e\left(\frac{26}{41}\right)\) |
| \(\chi_{739}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{41}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{27}{41}\right)\) | \(e\left(\frac{39}{41}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{10}{41}\right)\) | \(e\left(\frac{22}{41}\right)\) |
| \(\chi_{739}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{16}{41}\right)\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{25}{41}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{39}{41}\right)\) |
| \(\chi_{739}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{41}\right)\) | \(e\left(\frac{27}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{25}{41}\right)\) | \(e\left(\frac{34}{41}\right)\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{39}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) |
| \(\chi_{739}(125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{41}\right)\) | \(e\left(\frac{34}{41}\right)\) | \(e\left(\frac{25}{41}\right)\) | \(e\left(\frac{7}{41}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{33}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{27}{41}\right)\) | \(e\left(\frac{40}{41}\right)\) | \(e\left(\frac{6}{41}\right)\) |
| \(\chi_{739}(130,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{41}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{23}{41}\right)\) | \(e\left(\frac{28}{41}\right)\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{34}{41}\right)\) |
| \(\chi_{739}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{2}{41}\right)\) | \(e\left(\frac{28}{41}\right)\) | \(e\left(\frac{39}{41}\right)\) | \(e\left(\frac{16}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{10}{41}\right)\) |
| \(\chi_{739}(151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{28}{41}\right)\) | \(e\left(\frac{22}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{27}{41}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{24}{41}\right)\) |
| \(\chi_{739}(191,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{24}{41}\right)\) | \(e\left(\frac{10}{41}\right)\) | \(e\left(\frac{2}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{22}{41}\right)\) | \(e\left(\frac{32}{41}\right)\) |
| \(\chi_{739}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{24}{41}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{28}{41}\right)\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{7}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{38}{41}\right)\) |
| \(\chi_{739}(270,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{7}{41}\right)\) | \(e\left(\frac{20}{41}\right)\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{24}{41}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) | \(e\left(\frac{23}{41}\right)\) |
| \(\chi_{739}(277,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{33}{41}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{25}{41}\right)\) | \(e\left(\frac{34}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) |
| \(\chi_{739}(283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{39}{41}\right)\) | \(e\left(\frac{33}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) |
| \(\chi_{739}(293,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{33}{41}\right)\) | \(e\left(\frac{24}{41}\right)\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{34}{41}\right)\) | \(e\left(\frac{20}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) |
| \(\chi_{739}(367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{16}{41}\right)\) |
| \(\chi_{739}(376,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{2}{41}\right)\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{24}{41}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{19}{41}\right)\) |
| \(\chi_{739}(383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{41}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{23}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{8}{41}\right)\) |
| \(\chi_{739}(400,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{2}{41}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{7}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{30}{41}\right)\) |
| \(\chi_{739}(401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{10}{41}\right)\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{16}{41}\right)\) | \(e\left(\frac{23}{41}\right)\) | \(e\left(\frac{28}{41}\right)\) | \(e\left(\frac{37}{41}\right)\) |
| \(\chi_{739}(414,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{10}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{39}{41}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{20}{41}\right)\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) |
| \(\chi_{739}(416,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{28}{41}\right)\) | \(e\left(\frac{23}{41}\right)\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) |
| \(\chi_{739}(438,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{23}{41}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{20}{41}\right)\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{33}{41}\right)\) |
| \(\chi_{739}(443,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{40}{41}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{23}{41}\right)\) | \(e\left(\frac{10}{41}\right)\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{25}{41}\right)\) |
| \(\chi_{739}(474,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{22}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{28}{41}\right)\) |
| \(\chi_{739}(478,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{40}{41}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{7}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{2}{41}\right)\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{5}{41}\right)\) |
| \(\chi_{739}(487,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{41}\right)\) | \(e\left(\frac{40}{41}\right)\) | \(e\left(\frac{27}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{33}{41}\right)\) | \(e\left(\frac{34}{41}\right)\) | \(e\left(\frac{20}{41}\right)\) | \(e\left(\frac{39}{41}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{36}{41}\right)\) |
| \(\chi_{739}(495,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{27}{41}\right)\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{16}{41}\right)\) | \(e\left(\frac{7}{41}\right)\) | \(e\left(\frac{28}{41}\right)\) | \(e\left(\frac{2}{41}\right)\) | \(e\left(\frac{29}{41}\right)\) |
| \(\chi_{739}(538,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{7}{41}\right)\) | \(e\left(\frac{16}{41}\right)\) | \(e\left(\frac{34}{41}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{24}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{35}{41}\right)\) |
| \(\chi_{739}(541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{41}\right)\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{20}{41}\right)\) | \(e\left(\frac{22}{41}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{10}{41}\right)\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{13}{41}\right)\) |
| \(\chi_{739}(579,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{22}{41}\right)\) | \(e\left(\frac{25}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) |