from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4224, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([0,75,40,24]))
chi.galois_orbit()
[g,chi] = znchar(Mod(41,4224))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(4224\) | |
Conductor: | \(2112\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(80\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 2112.db | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{4224}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{80}\right)\) |
\(\chi_{4224}(233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{33}{80}\right)\) |
\(\chi_{4224}(281,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{80}\right)\) |
\(\chi_{4224}(425,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{41}{80}\right)\) |
\(\chi_{4224}(569,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{59}{80}\right)\) |
\(\chi_{4224}(761,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{80}\right)\) |
\(\chi_{4224}(809,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{57}{80}\right)\) |
\(\chi_{4224}(953,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{80}\right)\) |
\(\chi_{4224}(1097,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{29}{80}\right)\) |
\(\chi_{4224}(1289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{53}{80}\right)\) |
\(\chi_{4224}(1337,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{27}{80}\right)\) |
\(\chi_{4224}(1481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{61}{80}\right)\) |
\(\chi_{4224}(1625,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{79}{80}\right)\) |
\(\chi_{4224}(1817,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{23}{80}\right)\) |
\(\chi_{4224}(1865,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{77}{80}\right)\) |
\(\chi_{4224}(2009,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{31}{80}\right)\) |
\(\chi_{4224}(2153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{49}{80}\right)\) |
\(\chi_{4224}(2345,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{73}{80}\right)\) |
\(\chi_{4224}(2393,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{47}{80}\right)\) |
\(\chi_{4224}(2537,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{80}\right)\) |
\(\chi_{4224}(2681,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{80}\right)\) |
\(\chi_{4224}(2873,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{43}{80}\right)\) |
\(\chi_{4224}(2921,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{80}\right)\) |
\(\chi_{4224}(3065,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{51}{80}\right)\) |
\(\chi_{4224}(3209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{69}{80}\right)\) |
\(\chi_{4224}(3401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{80}\right)\) |
\(\chi_{4224}(3449,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{67}{80}\right)\) |
\(\chi_{4224}(3593,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{21}{80}\right)\) |
\(\chi_{4224}(3737,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{39}{80}\right)\) |
\(\chi_{4224}(3929,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{63}{80}\right)\) |
\(\chi_{4224}(3977,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{80}\right)\) |