from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2601, base_ring=CyclotomicField(408))
M = H._module
chi = DirichletCharacter(H, M([68,285]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,2601))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(2601\) | |
Conductor: | \(2601\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(408\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
Field of values: | $\Q(\zeta_{408})$ |
Fixed field: | Number field defined by a degree 408 polynomial (not computed) |
First 31 of 128 characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{2601}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{204}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{325}{408}\right)\) | \(e\left(\frac{383}{408}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{95}{408}\right)\) | \(e\left(\frac{25}{102}\right)\) | \(e\left(\frac{337}{408}\right)\) | \(e\left(\frac{28}{51}\right)\) |
\(\chi_{2601}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{204}\right)\) | \(e\left(\frac{89}{102}\right)\) | \(e\left(\frac{401}{408}\right)\) | \(e\left(\frac{283}{408}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{57}{136}\right)\) | \(e\left(\frac{67}{408}\right)\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{53}{408}\right)\) | \(e\left(\frac{38}{51}\right)\) |
\(\chi_{2601}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{204}\right)\) | \(e\left(\frac{59}{102}\right)\) | \(e\left(\frac{71}{408}\right)\) | \(e\left(\frac{277}{408}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{63}{136}\right)\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{29}{102}\right)\) | \(e\left(\frac{395}{408}\right)\) | \(e\left(\frac{8}{51}\right)\) |
\(\chi_{2601}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{204}\right)\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{11}{408}\right)\) | \(e\left(\frac{313}{408}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{27}{136}\right)\) | \(e\left(\frac{361}{408}\right)\) | \(e\left(\frac{95}{102}\right)\) | \(e\left(\frac{383}{408}\right)\) | \(e\left(\frac{35}{51}\right)\) |
\(\chi_{2601}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{204}\right)\) | \(e\left(\frac{19}{102}\right)\) | \(e\left(\frac{73}{408}\right)\) | \(e\left(\frac{371}{408}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{105}{136}\right)\) | \(e\left(\frac{59}{408}\right)\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{205}{408}\right)\) | \(e\left(\frac{19}{51}\right)\) |
\(\chi_{2601}(104,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{204}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{29}{408}\right)\) | \(e\left(\frac{343}{408}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{133}{136}\right)\) | \(e\left(\frac{247}{408}\right)\) | \(e\left(\frac{65}{102}\right)\) | \(e\left(\frac{305}{408}\right)\) | \(e\left(\frac{32}{51}\right)\) |
\(\chi_{2601}(128,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{204}\right)\) | \(e\left(\frac{43}{102}\right)\) | \(e\left(\frac{235}{408}\right)\) | \(e\left(\frac{233}{408}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{107}{136}\right)\) | \(e\left(\frac{257}{408}\right)\) | \(e\left(\frac{73}{102}\right)\) | \(e\left(\frac{319}{408}\right)\) | \(e\left(\frac{43}{51}\right)\) |
\(\chi_{2601}(185,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{204}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{233}{408}\right)\) | \(e\left(\frac{139}{408}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{65}{136}\right)\) | \(e\left(\frac{43}{408}\right)\) | \(e\left(\frac{65}{102}\right)\) | \(e\left(\frac{101}{408}\right)\) | \(e\left(\frac{32}{51}\right)\) |
\(\chi_{2601}(212,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{204}\right)\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{215}{408}\right)\) | \(e\left(\frac{109}{408}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{95}{136}\right)\) | \(e\left(\frac{157}{408}\right)\) | \(e\left(\frac{95}{102}\right)\) | \(e\left(\frac{179}{408}\right)\) | \(e\left(\frac{35}{51}\right)\) |
\(\chi_{2601}(230,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{204}\right)\) | \(e\left(\frac{59}{102}\right)\) | \(e\left(\frac{275}{408}\right)\) | \(e\left(\frac{73}{408}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{131}{136}\right)\) | \(e\left(\frac{49}{408}\right)\) | \(e\left(\frac{29}{102}\right)\) | \(e\left(\frac{191}{408}\right)\) | \(e\left(\frac{8}{51}\right)\) |
\(\chi_{2601}(236,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{204}\right)\) | \(e\left(\frac{13}{102}\right)\) | \(e\left(\frac{313}{408}\right)\) | \(e\left(\frac{227}{408}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{35}{408}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{13}{51}\right)\) |
\(\chi_{2601}(257,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{204}\right)\) | \(e\left(\frac{89}{102}\right)\) | \(e\left(\frac{197}{408}\right)\) | \(e\left(\frac{79}{408}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{125}{136}\right)\) | \(e\left(\frac{271}{408}\right)\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{257}{408}\right)\) | \(e\left(\frac{38}{51}\right)\) |
\(\chi_{2601}(263,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{204}\right)\) | \(e\left(\frac{61}{102}\right)\) | \(e\left(\frac{127}{408}\right)\) | \(e\left(\frac{53}{408}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{125}{408}\right)\) | \(e\left(\frac{49}{102}\right)\) | \(e\left(\frac{379}{408}\right)\) | \(e\left(\frac{10}{51}\right)\) |
\(\chi_{2601}(281,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{204}\right)\) | \(e\left(\frac{67}{102}\right)\) | \(e\left(\frac{91}{408}\right)\) | \(e\left(\frac{401}{408}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{75}{136}\right)\) | \(e\left(\frac{353}{408}\right)\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{127}{408}\right)\) | \(e\left(\frac{16}{51}\right)\) |
\(\chi_{2601}(308,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{204}\right)\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{263}{408}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{77}{136}\right)\) | \(e\left(\frac{143}{408}\right)\) | \(e\left(\frac{43}{102}\right)\) | \(e\left(\frac{241}{408}\right)\) | \(e\left(\frac{40}{51}\right)\) |
\(\chi_{2601}(338,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{204}\right)\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{65}{408}\right)\) | \(e\left(\frac{403}{408}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{73}{136}\right)\) | \(e\left(\frac{19}{408}\right)\) | \(e\left(\frac{5}{102}\right)\) | \(e\left(\frac{149}{408}\right)\) | \(e\left(\frac{26}{51}\right)\) |
\(\chi_{2601}(365,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{204}\right)\) | \(e\left(\frac{11}{102}\right)\) | \(e\left(\frac{359}{408}\right)\) | \(e\left(\frac{349}{408}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{127}{136}\right)\) | \(e\left(\frac{61}{408}\right)\) | \(e\left(\frac{59}{102}\right)\) | \(e\left(\frac{371}{408}\right)\) | \(e\left(\frac{11}{51}\right)\) |
\(\chi_{2601}(383,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{204}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{131}{408}\right)\) | \(e\left(\frac{241}{408}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{99}{136}\right)\) | \(e\left(\frac{145}{408}\right)\) | \(e\left(\frac{65}{102}\right)\) | \(e\left(\frac{407}{408}\right)\) | \(e\left(\frac{32}{51}\right)\) |
\(\chi_{2601}(389,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{204}\right)\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{145}{408}\right)\) | \(e\left(\frac{83}{408}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{121}{136}\right)\) | \(e\left(\frac{11}{408}\right)\) | \(e\left(\frac{19}{102}\right)\) | \(e\left(\frac{301}{408}\right)\) | \(e\left(\frac{7}{51}\right)\) |
\(\chi_{2601}(410,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{204}\right)\) | \(e\left(\frac{95}{102}\right)\) | \(e\left(\frac{365}{408}\right)\) | \(e\left(\frac{223}{408}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{117}{136}\right)\) | \(e\left(\frac{295}{408}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{209}{408}\right)\) | \(e\left(\frac{44}{51}\right)\) |
\(\chi_{2601}(416,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{204}\right)\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{271}{408}\right)\) | \(e\left(\frac{293}{408}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{47}{136}\right)\) | \(e\left(\frac{29}{408}\right)\) | \(e\left(\frac{13}{102}\right)\) | \(e\left(\frac{163}{408}\right)\) | \(e\left(\frac{37}{51}\right)\) |
\(\chi_{2601}(434,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{204}\right)\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{355}{408}\right)\) | \(e\left(\frac{161}{408}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{43}{136}\right)\) | \(e\left(\frac{41}{408}\right)\) | \(e\left(\frac{43}{102}\right)\) | \(e\left(\frac{343}{408}\right)\) | \(e\left(\frac{40}{51}\right)\) |
\(\chi_{2601}(461,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{204}\right)\) | \(e\left(\frac{97}{102}\right)\) | \(e\left(\frac{13}{408}\right)\) | \(e\left(\frac{407}{408}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{69}{136}\right)\) | \(e\left(\frac{167}{408}\right)\) | \(e\left(\frac{1}{102}\right)\) | \(e\left(\frac{193}{408}\right)\) | \(e\left(\frac{46}{51}\right)\) |
\(\chi_{2601}(491,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{204}\right)\) | \(e\left(\frac{71}{102}\right)\) | \(e\left(\frac{305}{408}\right)\) | \(e\left(\frac{259}{408}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{81}{136}\right)\) | \(e\left(\frac{403}{408}\right)\) | \(e\left(\frac{47}{102}\right)\) | \(e\left(\frac{197}{408}\right)\) | \(e\left(\frac{20}{51}\right)\) |
\(\chi_{2601}(518,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{204}\right)\) | \(e\left(\frac{89}{102}\right)\) | \(e\left(\frac{95}{408}\right)\) | \(e\left(\frac{181}{408}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{23}{136}\right)\) | \(e\left(\frac{373}{408}\right)\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{155}{408}\right)\) | \(e\left(\frac{38}{51}\right)\) |
\(\chi_{2601}(536,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{204}\right)\) | \(e\left(\frac{5}{102}\right)\) | \(e\left(\frac{395}{408}\right)\) | \(e\left(\frac{1}{408}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{67}{136}\right)\) | \(e\left(\frac{241}{408}\right)\) | \(e\left(\frac{101}{102}\right)\) | \(e\left(\frac{215}{408}\right)\) | \(e\left(\frac{5}{51}\right)\) |
\(\chi_{2601}(542,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{204}\right)\) | \(e\left(\frac{1}{102}\right)\) | \(e\left(\frac{385}{408}\right)\) | \(e\left(\frac{347}{408}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{129}{136}\right)\) | \(e\left(\frac{395}{408}\right)\) | \(e\left(\frac{61}{102}\right)\) | \(e\left(\frac{349}{408}\right)\) | \(e\left(\frac{1}{51}\right)\) |
\(\chi_{2601}(563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{204}\right)\) | \(e\left(\frac{101}{102}\right)\) | \(e\left(\frac{125}{408}\right)\) | \(e\left(\frac{367}{408}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{109}{136}\right)\) | \(e\left(\frac{319}{408}\right)\) | \(e\left(\frac{41}{102}\right)\) | \(e\left(\frac{161}{408}\right)\) | \(e\left(\frac{50}{51}\right)\) |
\(\chi_{2601}(569,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{204}\right)\) | \(e\left(\frac{13}{102}\right)\) | \(e\left(\frac{7}{408}\right)\) | \(e\left(\frac{125}{408}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{79}{136}\right)\) | \(e\left(\frac{341}{408}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{355}{408}\right)\) | \(e\left(\frac{13}{51}\right)\) |
\(\chi_{2601}(587,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{204}\right)\) | \(e\left(\frac{13}{102}\right)\) | \(e\left(\frac{211}{408}\right)\) | \(e\left(\frac{329}{408}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{11}{136}\right)\) | \(e\left(\frac{137}{408}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{151}{408}\right)\) | \(e\left(\frac{13}{51}\right)\) |
\(\chi_{2601}(614,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{204}\right)\) | \(e\left(\frac{1}{102}\right)\) | \(e\left(\frac{181}{408}\right)\) | \(e\left(\frac{143}{408}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{61}{136}\right)\) | \(e\left(\frac{191}{408}\right)\) | \(e\left(\frac{61}{102}\right)\) | \(e\left(\frac{145}{408}\right)\) | \(e\left(\frac{1}{51}\right)\) |