from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2025, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([25,81]))
chi.galois_orbit()
[g,chi] = znchar(Mod(44,2025))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(2025\) | |
Conductor: | \(675\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(90\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 675.bh | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | 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 90 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{2025}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) |
\(\chi_{2025}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) |
\(\chi_{2025}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{71}{90}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) |
\(\chi_{2025}(314,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{77}{90}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) |
\(\chi_{2025}(359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) |
\(\chi_{2025}(494,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) |
\(\chi_{2025}(584,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) |
\(\chi_{2025}(629,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) |
\(\chi_{2025}(719,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) |
\(\chi_{2025}(764,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) |
\(\chi_{2025}(854,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) |
\(\chi_{2025}(989,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) |
\(\chi_{2025}(1034,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) |
\(\chi_{2025}(1169,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{71}{90}\right)\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) |
\(\chi_{2025}(1259,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{83}{90}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) |
\(\chi_{2025}(1304,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) |
\(\chi_{2025}(1394,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{71}{90}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) |
\(\chi_{2025}(1439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{77}{90}\right)\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) |
\(\chi_{2025}(1529,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) |
\(\chi_{2025}(1664,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{83}{90}\right)\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) |
\(\chi_{2025}(1709,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{83}{90}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) |
\(\chi_{2025}(1844,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) |
\(\chi_{2025}(1934,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{77}{90}\right)\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) |
\(\chi_{2025}(1979,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) |