from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(121, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,121))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(121\) | |
| Conductor: | \(121\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(55\) | 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: | $\Q(\zeta_{55})$ |
| Fixed field: | Number field defined by a degree 55 polynomial |
First 31 of 40 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{121}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) |
| \(\chi_{121}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
| \(\chi_{121}(14,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
| \(\chi_{121}(15,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
| \(\chi_{121}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) |
| \(\chi_{121}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) |
| \(\chi_{121}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) |
| \(\chi_{121}(26,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) |
| \(\chi_{121}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) |
| \(\chi_{121}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) |
| \(\chi_{121}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) |
| \(\chi_{121}(38,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) |
| \(\chi_{121}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
| \(\chi_{121}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) |
| \(\chi_{121}(48,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) |
| \(\chi_{121}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) |
| \(\chi_{121}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) |
| \(\chi_{121}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) |
| \(\chi_{121}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) |
| \(\chi_{121}(60,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) |
| \(\chi_{121}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) |
| \(\chi_{121}(69,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) |
| \(\chi_{121}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) |
| \(\chi_{121}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) |
| \(\chi_{121}(75,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) |
| \(\chi_{121}(80,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) |
| \(\chi_{121}(82,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) |
| \(\chi_{121}(86,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) |
| \(\chi_{121}(91,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) |
| \(\chi_{121}(92,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) |
| \(\chi_{121}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) |