from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6001, base_ring=CyclotomicField(176))
M = H._module
chi = DirichletCharacter(H, M([0,83]))
chi.galois_orbit()
[g,chi] = znchar(Mod(18,6001))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(6001\) | |
Conductor: | \(353\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(176\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 353.k | 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: | $\Q(\zeta_{176})$ |
Fixed field: | Number field defined by a degree 176 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{6001}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{83}{176}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{127}{176}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{13}{44}\right)\) |
\(\chi_{6001}(86,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{101}{176}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{57}{176}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{13}{88}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{19}{44}\right)\) |
\(\chi_{6001}(120,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{49}{176}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{5}{176}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{31}{44}\right)\) |
\(\chi_{6001}(188,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{97}{176}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{53}{176}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{9}{88}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{3}{44}\right)\) |
\(\chi_{6001}(307,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{173}{176}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{129}{176}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{85}{88}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{43}{44}\right)\) |
\(\chi_{6001}(392,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{169}{176}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{125}{176}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{27}{44}\right)\) |
\(\chi_{6001}(562,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{65}{176}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{21}{176}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{65}{88}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{44}\right)\) |
\(\chi_{6001}(579,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{125}{176}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{81}{176}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{27}{44}\right)\) |
\(\chi_{6001}(613,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{135}{176}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{3}{176}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{44}\right)\) |
\(\chi_{6001}(630,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{111}{176}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{155}{176}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{37}{44}\right)\) |
\(\chi_{6001}(681,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{45}{176}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1}{176}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{15}{44}\right)\) |
\(\chi_{6001}(715,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{1}{176}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{133}{176}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{15}{44}\right)\) |
\(\chi_{6001}(749,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{19}{176}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{63}{176}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{19}{88}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{21}{44}\right)\) |
\(\chi_{6001}(800,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{15}{176}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{59}{176}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{15}{88}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{44}\right)\) |
\(\chi_{6001}(902,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{87}{176}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{131}{176}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{29}{44}\right)\) |
\(\chi_{6001}(987,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{159}{176}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{27}{176}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{9}{44}\right)\) |
\(\chi_{6001}(1021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{29}{176}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{161}{176}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{39}{44}\right)\) |
\(\chi_{6001}(1089,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{61}{176}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{17}{176}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{35}{44}\right)\) |
\(\chi_{6001}(1106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{109}{176}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{65}{176}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{21}{88}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{44}\right)\) |
\(\chi_{6001}(1157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{5}{176}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{137}{176}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{31}{44}\right)\) |
\(\chi_{6001}(1259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{27}{176}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{71}{176}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{27}{88}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{9}{44}\right)\) |
\(\chi_{6001}(1565,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{115}{176}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{159}{176}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{27}{88}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{9}{44}\right)\) |
\(\chi_{6001}(1667,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{93}{176}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{49}{176}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{31}{44}\right)\) |
\(\chi_{6001}(1718,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{21}{176}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{153}{176}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{21}{88}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{44}\right)\) |
\(\chi_{6001}(1735,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{149}{176}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{105}{176}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{35}{44}\right)\) |
\(\chi_{6001}(1803,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{117}{176}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{73}{176}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{39}{44}\right)\) |
\(\chi_{6001}(1837,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{71}{176}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{115}{176}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{9}{44}\right)\) |
\(\chi_{6001}(1922,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{175}{176}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{43}{176}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{29}{44}\right)\) |
\(\chi_{6001}(2024,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{103}{176}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{147}{176}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{15}{88}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{44}\right)\) |
\(\chi_{6001}(2075,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{107}{176}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{151}{176}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{19}{88}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{21}{44}\right)\) |
\(\chi_{6001}(2109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{89}{176}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{45}{176}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{15}{44}\right)\) |