from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4224, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,5,0,16]))
chi.galois_orbit()
[g,chi] = znchar(Mod(103,4224))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4224\) | |
| Conductor: | \(704\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(80\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 704.bl | 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: | $\Q(\zeta_{80})$ |
| Fixed field: | Number field defined by a degree 80 polynomial |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4224}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{31}{80}\right)\) |
| \(\chi_{4224}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{77}{80}\right)\) |
| \(\chi_{4224}(295,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{80}\right)\) |
| \(\chi_{4224}(487,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{79}{80}\right)\) |
| \(\chi_{4224}(631,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{61}{80}\right)\) |
| \(\chi_{4224}(775,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{27}{80}\right)\) |
| \(\chi_{4224}(823,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{53}{80}\right)\) |
| \(\chi_{4224}(1015,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{29}{80}\right)\) |
| \(\chi_{4224}(1159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{80}\right)\) |
| \(\chi_{4224}(1303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{57}{80}\right)\) |
| \(\chi_{4224}(1351,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{80}\right)\) |
| \(\chi_{4224}(1543,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{59}{80}\right)\) |
| \(\chi_{4224}(1687,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{41}{80}\right)\) |
| \(\chi_{4224}(1831,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{80}\right)\) |
| \(\chi_{4224}(1879,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{33}{80}\right)\) |
| \(\chi_{4224}(2071,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{80}\right)\) |
| \(\chi_{4224}(2215,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{71}{80}\right)\) |
| \(\chi_{4224}(2359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{37}{80}\right)\) |
| \(\chi_{4224}(2407,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{63}{80}\right)\) |
| \(\chi_{4224}(2599,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{39}{80}\right)\) |
| \(\chi_{4224}(2743,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{21}{80}\right)\) |
| \(\chi_{4224}(2887,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{67}{80}\right)\) |
| \(\chi_{4224}(2935,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{80}\right)\) |
| \(\chi_{4224}(3127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{69}{80}\right)\) |
| \(\chi_{4224}(3271,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{51}{80}\right)\) |
| \(\chi_{4224}(3415,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{80}\right)\) |
| \(\chi_{4224}(3463,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{43}{80}\right)\) |
| \(\chi_{4224}(3655,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{80}\right)\) |
| \(\chi_{4224}(3799,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{80}\right)\) |
| \(\chi_{4224}(3943,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{47}{80}\right)\) |
| \(\chi_{4224}(3991,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{73}{80}\right)\) |