from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([0,36,20]))
chi.galois_orbit()
[g,chi] = znchar(Mod(16,1045))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(1045\) | |
Conductor: | \(209\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(45\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 209.u | 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: | 45.45.43679806300610465846484971330073185012597520004657724600953543350870941304329239684756561.1 |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{1045}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) |
\(\chi_{1045}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) |
\(\chi_{1045}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) |
\(\chi_{1045}(196,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) |
\(\chi_{1045}(251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) |
\(\chi_{1045}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) |
\(\chi_{1045}(291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) |
\(\chi_{1045}(301,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) |
\(\chi_{1045}(346,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) |
\(\chi_{1045}(366,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) |
\(\chi_{1045}(416,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) |
\(\chi_{1045}(511,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) |
\(\chi_{1045}(576,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) |
\(\chi_{1045}(586,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) |
\(\chi_{1045}(631,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) |
\(\chi_{1045}(636,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) |
\(\chi_{1045}(731,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) |
\(\chi_{1045}(746,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) |
\(\chi_{1045}(796,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) |
\(\chi_{1045}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) |
\(\chi_{1045}(861,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) |
\(\chi_{1045}(916,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) |
\(\chi_{1045}(966,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) |
\(\chi_{1045}(1016,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) |