from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,363))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(363\) | |
| Conductor: | \(121\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(55\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 121.g | 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{363}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) |
| \(\chi_{363}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) |
| \(\chi_{363}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) |
| \(\chi_{363}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) |
| \(\chi_{363}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) |
| \(\chi_{363}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) |
| \(\chi_{363}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) |
| \(\chi_{363}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) |
| \(\chi_{363}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) |
| \(\chi_{363}(82,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) |
| \(\chi_{363}(91,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) |
| \(\chi_{363}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) |
| \(\chi_{363}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) |
| \(\chi_{363}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) |
| \(\chi_{363}(136,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) |
| \(\chi_{363}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) |
| \(\chi_{363}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) |
| \(\chi_{363}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) |
| \(\chi_{363}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) |
| \(\chi_{363}(190,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) |
| \(\chi_{363}(196,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) |
| \(\chi_{363}(214,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) |
| \(\chi_{363}(223,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) |
| \(\chi_{363}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) |
| \(\chi_{363}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) |
| \(\chi_{363}(247,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) |
| \(\chi_{363}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) |
| \(\chi_{363}(262,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) |
| \(\chi_{363}(268,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) |
| \(\chi_{363}(280,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) |
| \(\chi_{363}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) |