from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(107, base_ring=CyclotomicField(106))
M = H._module
chi = DirichletCharacter(H, M([70]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,107))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(107\) | |
| Conductor: | \(107\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(53\) | 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: | $\Q(\zeta_{53})$ |
| Fixed field: | Number field defined by a degree 53 polynomial |
First 31 of 52 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{107}(3,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{53}\right)\) | \(e\left(\frac{12}{53}\right)\) | \(e\left(\frac{17}{53}\right)\) | \(e\left(\frac{2}{53}\right)\) | \(e\left(\frac{47}{53}\right)\) | \(e\left(\frac{21}{53}\right)\) | \(e\left(\frac{52}{53}\right)\) | \(e\left(\frac{24}{53}\right)\) | \(e\left(\frac{37}{53}\right)\) | \(e\left(\frac{28}{53}\right)\) |
| \(\chi_{107}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{53}\right)\) | \(e\left(\frac{17}{53}\right)\) | \(e\left(\frac{2}{53}\right)\) | \(e\left(\frac{47}{53}\right)\) | \(e\left(\frac{18}{53}\right)\) | \(e\left(\frac{43}{53}\right)\) | \(e\left(\frac{3}{53}\right)\) | \(e\left(\frac{34}{53}\right)\) | \(e\left(\frac{48}{53}\right)\) | \(e\left(\frac{22}{53}\right)\) |
| \(\chi_{107}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{53}\right)\) | \(e\left(\frac{24}{53}\right)\) | \(e\left(\frac{34}{53}\right)\) | \(e\left(\frac{4}{53}\right)\) | \(e\left(\frac{41}{53}\right)\) | \(e\left(\frac{42}{53}\right)\) | \(e\left(\frac{51}{53}\right)\) | \(e\left(\frac{48}{53}\right)\) | \(e\left(\frac{21}{53}\right)\) | \(e\left(\frac{3}{53}\right)\) |
| \(\chi_{107}(10,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{53}\right)\) | \(e\left(\frac{37}{53}\right)\) | \(e\left(\frac{48}{53}\right)\) | \(e\left(\frac{15}{53}\right)\) | \(e\left(\frac{8}{53}\right)\) | \(e\left(\frac{25}{53}\right)\) | \(e\left(\frac{19}{53}\right)\) | \(e\left(\frac{21}{53}\right)\) | \(e\left(\frac{39}{53}\right)\) | \(e\left(\frac{51}{53}\right)\) |
| \(\chi_{107}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{53}\right)\) | \(e\left(\frac{28}{53}\right)\) | \(e\left(\frac{22}{53}\right)\) | \(e\left(\frac{40}{53}\right)\) | \(e\left(\frac{39}{53}\right)\) | \(e\left(\frac{49}{53}\right)\) | \(e\left(\frac{33}{53}\right)\) | \(e\left(\frac{3}{53}\right)\) | \(e\left(\frac{51}{53}\right)\) | \(e\left(\frac{30}{53}\right)\) |
| \(\chi_{107}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{53}\right)\) | \(e\left(\frac{29}{53}\right)\) | \(e\left(\frac{19}{53}\right)\) | \(e\left(\frac{49}{53}\right)\) | \(e\left(\frac{12}{53}\right)\) | \(e\left(\frac{11}{53}\right)\) | \(e\left(\frac{2}{53}\right)\) | \(e\left(\frac{5}{53}\right)\) | \(e\left(\frac{32}{53}\right)\) | \(e\left(\frac{50}{53}\right)\) |
| \(\chi_{107}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{53}\right)\) | \(e\left(\frac{13}{53}\right)\) | \(e\left(\frac{14}{53}\right)\) | \(e\left(\frac{11}{53}\right)\) | \(e\left(\frac{20}{53}\right)\) | \(e\left(\frac{36}{53}\right)\) | \(e\left(\frac{21}{53}\right)\) | \(e\left(\frac{26}{53}\right)\) | \(e\left(\frac{18}{53}\right)\) | \(e\left(\frac{48}{53}\right)\) |
| \(\chi_{107}(14,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{53}\right)\) | \(e\left(\frac{3}{53}\right)\) | \(e\left(\frac{44}{53}\right)\) | \(e\left(\frac{27}{53}\right)\) | \(e\left(\frac{25}{53}\right)\) | \(e\left(\frac{45}{53}\right)\) | \(e\left(\frac{13}{53}\right)\) | \(e\left(\frac{6}{53}\right)\) | \(e\left(\frac{49}{53}\right)\) | \(e\left(\frac{7}{53}\right)\) |
| \(\chi_{107}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{53}\right)\) | \(e\left(\frac{34}{53}\right)\) | \(e\left(\frac{4}{53}\right)\) | \(e\left(\frac{41}{53}\right)\) | \(e\left(\frac{36}{53}\right)\) | \(e\left(\frac{33}{53}\right)\) | \(e\left(\frac{6}{53}\right)\) | \(e\left(\frac{15}{53}\right)\) | \(e\left(\frac{43}{53}\right)\) | \(e\left(\frac{44}{53}\right)\) |
| \(\chi_{107}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{53}\right)\) | \(e\left(\frac{27}{53}\right)\) | \(e\left(\frac{25}{53}\right)\) | \(e\left(\frac{31}{53}\right)\) | \(e\left(\frac{13}{53}\right)\) | \(e\left(\frac{34}{53}\right)\) | \(e\left(\frac{11}{53}\right)\) | \(e\left(\frac{1}{53}\right)\) | \(e\left(\frac{17}{53}\right)\) | \(e\left(\frac{10}{53}\right)\) |
| \(\chi_{107}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{53}\right)\) | \(e\left(\frac{50}{53}\right)\) | \(e\left(\frac{9}{53}\right)\) | \(e\left(\frac{26}{53}\right)\) | \(e\left(\frac{28}{53}\right)\) | \(e\left(\frac{8}{53}\right)\) | \(e\left(\frac{40}{53}\right)\) | \(e\left(\frac{47}{53}\right)\) | \(e\left(\frac{4}{53}\right)\) | \(e\left(\frac{46}{53}\right)\) |
| \(\chi_{107}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{53}\right)\) | \(e\left(\frac{4}{53}\right)\) | \(e\left(\frac{41}{53}\right)\) | \(e\left(\frac{36}{53}\right)\) | \(e\left(\frac{51}{53}\right)\) | \(e\left(\frac{7}{53}\right)\) | \(e\left(\frac{35}{53}\right)\) | \(e\left(\frac{8}{53}\right)\) | \(e\left(\frac{30}{53}\right)\) | \(e\left(\frac{27}{53}\right)\) |
| \(\chi_{107}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{53}\right)\) | \(e\left(\frac{36}{53}\right)\) | \(e\left(\frac{51}{53}\right)\) | \(e\left(\frac{6}{53}\right)\) | \(e\left(\frac{35}{53}\right)\) | \(e\left(\frac{10}{53}\right)\) | \(e\left(\frac{50}{53}\right)\) | \(e\left(\frac{19}{53}\right)\) | \(e\left(\frac{5}{53}\right)\) | \(e\left(\frac{31}{53}\right)\) |
| \(\chi_{107}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{53}\right)\) | \(e\left(\frac{7}{53}\right)\) | \(e\left(\frac{32}{53}\right)\) | \(e\left(\frac{10}{53}\right)\) | \(e\left(\frac{23}{53}\right)\) | \(e\left(\frac{52}{53}\right)\) | \(e\left(\frac{48}{53}\right)\) | \(e\left(\frac{14}{53}\right)\) | \(e\left(\frac{26}{53}\right)\) | \(e\left(\frac{34}{53}\right)\) |
| \(\chi_{107}(30,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{53}\right)\) | \(e\left(\frac{49}{53}\right)\) | \(e\left(\frac{12}{53}\right)\) | \(e\left(\frac{17}{53}\right)\) | \(e\left(\frac{2}{53}\right)\) | \(e\left(\frac{46}{53}\right)\) | \(e\left(\frac{18}{53}\right)\) | \(e\left(\frac{45}{53}\right)\) | \(e\left(\frac{23}{53}\right)\) | \(e\left(\frac{26}{53}\right)\) |
| \(\chi_{107}(33,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{53}\right)\) | \(e\left(\frac{40}{53}\right)\) | \(e\left(\frac{39}{53}\right)\) | \(e\left(\frac{42}{53}\right)\) | \(e\left(\frac{33}{53}\right)\) | \(e\left(\frac{17}{53}\right)\) | \(e\left(\frac{32}{53}\right)\) | \(e\left(\frac{27}{53}\right)\) | \(e\left(\frac{35}{53}\right)\) | \(e\left(\frac{5}{53}\right)\) |
| \(\chi_{107}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{53}\right)\) | \(e\left(\frac{43}{53}\right)\) | \(e\left(\frac{30}{53}\right)\) | \(e\left(\frac{16}{53}\right)\) | \(e\left(\frac{5}{53}\right)\) | \(e\left(\frac{9}{53}\right)\) | \(e\left(\frac{45}{53}\right)\) | \(e\left(\frac{33}{53}\right)\) | \(e\left(\frac{31}{53}\right)\) | \(e\left(\frac{12}{53}\right)\) |
| \(\chi_{107}(35,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{53}\right)\) | \(e\left(\frac{23}{53}\right)\) | \(e\left(\frac{37}{53}\right)\) | \(e\left(\frac{48}{53}\right)\) | \(e\left(\frac{15}{53}\right)\) | \(e\left(\frac{27}{53}\right)\) | \(e\left(\frac{29}{53}\right)\) | \(e\left(\frac{46}{53}\right)\) | \(e\left(\frac{40}{53}\right)\) | \(e\left(\frac{36}{53}\right)\) |
| \(\chi_{107}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{53}\right)\) | \(e\left(\frac{41}{53}\right)\) | \(e\left(\frac{36}{53}\right)\) | \(e\left(\frac{51}{53}\right)\) | \(e\left(\frac{6}{53}\right)\) | \(e\left(\frac{32}{53}\right)\) | \(e\left(\frac{1}{53}\right)\) | \(e\left(\frac{29}{53}\right)\) | \(e\left(\frac{16}{53}\right)\) | \(e\left(\frac{25}{53}\right)\) |
| \(\chi_{107}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{53}\right)\) | \(e\left(\frac{5}{53}\right)\) | \(e\left(\frac{38}{53}\right)\) | \(e\left(\frac{45}{53}\right)\) | \(e\left(\frac{24}{53}\right)\) | \(e\left(\frac{22}{53}\right)\) | \(e\left(\frac{4}{53}\right)\) | \(e\left(\frac{10}{53}\right)\) | \(e\left(\frac{11}{53}\right)\) | \(e\left(\frac{47}{53}\right)\) |
| \(\chi_{107}(39,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{53}\right)\) | \(e\left(\frac{25}{53}\right)\) | \(e\left(\frac{31}{53}\right)\) | \(e\left(\frac{13}{53}\right)\) | \(e\left(\frac{14}{53}\right)\) | \(e\left(\frac{4}{53}\right)\) | \(e\left(\frac{20}{53}\right)\) | \(e\left(\frac{50}{53}\right)\) | \(e\left(\frac{2}{53}\right)\) | \(e\left(\frac{23}{53}\right)\) |
| \(\chi_{107}(40,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{53}\right)\) | \(e\left(\frac{1}{53}\right)\) | \(e\left(\frac{50}{53}\right)\) | \(e\left(\frac{9}{53}\right)\) | \(e\left(\frac{26}{53}\right)\) | \(e\left(\frac{15}{53}\right)\) | \(e\left(\frac{22}{53}\right)\) | \(e\left(\frac{2}{53}\right)\) | \(e\left(\frac{34}{53}\right)\) | \(e\left(\frac{20}{53}\right)\) |
| \(\chi_{107}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{53}\right)\) | \(e\left(\frac{22}{53}\right)\) | \(e\left(\frac{40}{53}\right)\) | \(e\left(\frac{39}{53}\right)\) | \(e\left(\frac{42}{53}\right)\) | \(e\left(\frac{12}{53}\right)\) | \(e\left(\frac{7}{53}\right)\) | \(e\left(\frac{44}{53}\right)\) | \(e\left(\frac{6}{53}\right)\) | \(e\left(\frac{16}{53}\right)\) |
| \(\chi_{107}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{53}\right)\) | \(e\left(\frac{15}{53}\right)\) | \(e\left(\frac{8}{53}\right)\) | \(e\left(\frac{29}{53}\right)\) | \(e\left(\frac{19}{53}\right)\) | \(e\left(\frac{13}{53}\right)\) | \(e\left(\frac{12}{53}\right)\) | \(e\left(\frac{30}{53}\right)\) | \(e\left(\frac{33}{53}\right)\) | \(e\left(\frac{35}{53}\right)\) |
| \(\chi_{107}(44,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{53}\right)\) | \(e\left(\frac{45}{53}\right)\) | \(e\left(\frac{24}{53}\right)\) | \(e\left(\frac{34}{53}\right)\) | \(e\left(\frac{4}{53}\right)\) | \(e\left(\frac{39}{53}\right)\) | \(e\left(\frac{36}{53}\right)\) | \(e\left(\frac{37}{53}\right)\) | \(e\left(\frac{46}{53}\right)\) | \(e\left(\frac{52}{53}\right)\) |
| \(\chi_{107}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{53}\right)\) | \(e\left(\frac{31}{53}\right)\) | \(e\left(\frac{13}{53}\right)\) | \(e\left(\frac{14}{53}\right)\) | \(e\left(\frac{11}{53}\right)\) | \(e\left(\frac{41}{53}\right)\) | \(e\left(\frac{46}{53}\right)\) | \(e\left(\frac{9}{53}\right)\) | \(e\left(\frac{47}{53}\right)\) | \(e\left(\frac{37}{53}\right)\) |
| \(\chi_{107}(48,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{53}\right)\) | \(e\left(\frac{46}{53}\right)\) | \(e\left(\frac{21}{53}\right)\) | \(e\left(\frac{43}{53}\right)\) | \(e\left(\frac{30}{53}\right)\) | \(e\left(\frac{1}{53}\right)\) | \(e\left(\frac{5}{53}\right)\) | \(e\left(\frac{39}{53}\right)\) | \(e\left(\frac{27}{53}\right)\) | \(e\left(\frac{19}{53}\right)\) |
| \(\chi_{107}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{53}\right)\) | \(e\left(\frac{42}{53}\right)\) | \(e\left(\frac{33}{53}\right)\) | \(e\left(\frac{7}{53}\right)\) | \(e\left(\frac{32}{53}\right)\) | \(e\left(\frac{47}{53}\right)\) | \(e\left(\frac{23}{53}\right)\) | \(e\left(\frac{31}{53}\right)\) | \(e\left(\frac{50}{53}\right)\) | \(e\left(\frac{45}{53}\right)\) |
| \(\chi_{107}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{53}\right)\) | \(e\left(\frac{30}{53}\right)\) | \(e\left(\frac{16}{53}\right)\) | \(e\left(\frac{5}{53}\right)\) | \(e\left(\frac{38}{53}\right)\) | \(e\left(\frac{26}{53}\right)\) | \(e\left(\frac{24}{53}\right)\) | \(e\left(\frac{7}{53}\right)\) | \(e\left(\frac{13}{53}\right)\) | \(e\left(\frac{17}{53}\right)\) |
| \(\chi_{107}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{53}\right)\) | \(e\left(\frac{18}{53}\right)\) | \(e\left(\frac{52}{53}\right)\) | \(e\left(\frac{3}{53}\right)\) | \(e\left(\frac{44}{53}\right)\) | \(e\left(\frac{5}{53}\right)\) | \(e\left(\frac{25}{53}\right)\) | \(e\left(\frac{36}{53}\right)\) | \(e\left(\frac{29}{53}\right)\) | \(e\left(\frac{42}{53}\right)\) |
| \(\chi_{107}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{53}\right)\) | \(e\left(\frac{20}{53}\right)\) | \(e\left(\frac{46}{53}\right)\) | \(e\left(\frac{21}{53}\right)\) | \(e\left(\frac{43}{53}\right)\) | \(e\left(\frac{35}{53}\right)\) | \(e\left(\frac{16}{53}\right)\) | \(e\left(\frac{40}{53}\right)\) | \(e\left(\frac{44}{53}\right)\) | \(e\left(\frac{29}{53}\right)\) |