from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(413, base_ring=CyclotomicField(58))
M = H._module
chi = DirichletCharacter(H, M([29,51]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,413))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(413\) | |
| Conductor: | \(413\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(58\) | 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: | $\Q(\zeta_{29})$ |
| Fixed field: | Number field defined by a degree 58 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{413}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{13}{58}\right)\) |
| \(\chi_{413}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{49}{58}\right)\) |
| \(\chi_{413}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{15}{58}\right)\) |
| \(\chi_{413}(55,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{17}{58}\right)\) |
| \(\chi_{413}(69,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{45}{58}\right)\) |
| \(\chi_{413}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{1}{58}\right)\) |
| \(\chi_{413}(90,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{25}{58}\right)\) |
| \(\chi_{413}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{27}{58}\right)\) |
| \(\chi_{413}(111,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{37}{58}\right)\) |
| \(\chi_{413}(132,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{31}{58}\right)\) |
| \(\chi_{413}(160,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{21}{58}\right)\) |
| \(\chi_{413}(174,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{19}{58}\right)\) |
| \(\chi_{413}(188,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{53}{58}\right)\) |
| \(\chi_{413}(195,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{3}{58}\right)\) |
| \(\chi_{413}(209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{57}{58}\right)\) |
| \(\chi_{413}(216,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{39}{58}\right)\) |
| \(\chi_{413}(244,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{11}{58}\right)\) |
| \(\chi_{413}(279,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{5}{58}\right)\) |
| \(\chi_{413}(286,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{9}{58}\right)\) |
| \(\chi_{413}(328,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{43}{58}\right)\) |
| \(\chi_{413}(335,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{33}{58}\right)\) |
| \(\chi_{413}(342,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{7}{58}\right)\) |
| \(\chi_{413}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{51}{58}\right)\) |
| \(\chi_{413}(356,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{23}{58}\right)\) |
| \(\chi_{413}(377,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{55}{58}\right)\) |
| \(\chi_{413}(384,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{35}{58}\right)\) |
| \(\chi_{413}(391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{47}{58}\right)\) |
| \(\chi_{413}(398,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{41}{58}\right)\) |