from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2850, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([0,72,10]))
chi.galois_orbit()
[g,chi] = znchar(Mod(61,2850))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(2850\) | |
Conductor: | \(475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(45\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 475.bc | 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: | $\Q(\zeta_{45})$ |
Fixed field: | Number field defined by a degree 45 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{2850}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{7}{9}\right)\) |
\(\chi_{2850}(271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{2850}(481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{2850}(511,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{2850}(541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{2850}(631,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{7}{9}\right)\) |
\(\chi_{2850}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{2850}(871,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{5}{9}\right)\) |
\(\chi_{2850}(1081,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{2850}(1111,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{2850}(1411,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{2850}(1441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{5}{9}\right)\) |
\(\chi_{2850}(1621,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{2850}(1681,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{2850}(1771,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{7}{9}\right)\) |
\(\chi_{2850}(1981,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{2850}(2011,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{5}{9}\right)\) |
\(\chi_{2850}(2191,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{2850}(2221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{2850}(2341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{7}{9}\right)\) |
\(\chi_{2850}(2581,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{5}{9}\right)\) |
\(\chi_{2850}(2761,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{2850}(2791,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{2850}(2821,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{1}{9}\right)\) |