from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1156, base_ring=CyclotomicField(136))
M = H._module
chi = DirichletCharacter(H, M([68,115]))
chi.galois_orbit()
[g,chi] = znchar(Mod(15,1156))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(1156\) | |
Conductor: | \(1156\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(136\) | 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: | $\Q(\zeta_{136})$ |
Fixed field: | Number field defined by a degree 136 polynomial (not computed) |
First 31 of 64 characters in Galois orbit
Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{1156}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{136}\right)\) | \(e\left(\frac{87}{136}\right)\) | \(e\left(\frac{77}{136}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{129}{136}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{9}{136}\right)\) |
\(\chi_{1156}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{75}{136}\right)\) | \(e\left(\frac{107}{136}\right)\) | \(e\left(\frac{65}{136}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{133}{136}\right)\) |
\(\chi_{1156}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{136}\right)\) | \(e\left(\frac{37}{136}\right)\) | \(e\left(\frac{39}{136}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{83}{136}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{107}{136}\right)\) |
\(\chi_{1156}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{136}\right)\) | \(e\left(\frac{1}{136}\right)\) | \(e\left(\frac{115}{136}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{39}{136}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{47}{136}\right)\) |
\(\chi_{1156}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{136}\right)\) | \(e\left(\frac{47}{136}\right)\) | \(e\left(\frac{101}{136}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{65}{136}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{33}{136}\right)\) |
\(\chi_{1156}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{136}\right)\) | \(e\left(\frac{11}{136}\right)\) | \(e\left(\frac{41}{136}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{21}{136}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{109}{136}\right)\) |
\(\chi_{1156}(111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{136}\right)\) | \(e\left(\frac{61}{136}\right)\) | \(e\left(\frac{79}{136}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{67}{136}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{11}{136}\right)\) |
\(\chi_{1156}(127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{136}\right)\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{75}{136}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{55}{136}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{7}{136}\right)\) |
\(\chi_{1156}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{136}\right)\) | \(e\left(\frac{7}{136}\right)\) | \(e\left(\frac{125}{136}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{1}{136}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{57}{136}\right)\) |
\(\chi_{1156}(195,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{136}\right)\) | \(e\left(\frac{89}{136}\right)\) | \(e\left(\frac{35}{136}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{71}{136}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{103}{136}\right)\) |
\(\chi_{1156}(219,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{103}{136}\right)\) | \(e\left(\frac{13}{136}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{73}{136}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{81}{136}\right)\) |
\(\chi_{1156}(223,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{136}\right)\) | \(e\left(\frac{91}{136}\right)\) | \(e\left(\frac{129}{136}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{13}{136}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{61}{136}\right)\) |
\(\chi_{1156}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{136}\right)\) | \(e\left(\frac{109}{136}\right)\) | \(e\left(\frac{23}{136}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{35}{136}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{91}{136}\right)\) |
\(\chi_{1156}(263,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{136}\right)\) | \(e\left(\frac{65}{136}\right)\) | \(e\left(\frac{131}{136}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{87}{136}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{63}{136}\right)\) |
\(\chi_{1156}(287,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{136}\right)\) | \(e\left(\frac{63}{136}\right)\) | \(e\left(\frac{37}{136}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{9}{136}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{105}{136}\right)\) |
\(\chi_{1156}(291,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{136}\right)\) | \(e\left(\frac{131}{136}\right)\) | \(e\left(\frac{105}{136}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{77}{136}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{37}{136}\right)\) |
\(\chi_{1156}(315,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{136}\right)\) | \(e\left(\frac{133}{136}\right)\) | \(e\left(\frac{63}{136}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{19}{136}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{131}{136}\right)\) |
\(\chi_{1156}(331,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{105}{136}\right)\) | \(e\left(\frac{41}{136}\right)\) | \(e\left(\frac{91}{136}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{103}{136}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{23}{136}\right)\) |
\(\chi_{1156}(355,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{136}\right)\) | \(e\left(\frac{23}{136}\right)\) | \(e\left(\frac{61}{136}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{81}{136}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{129}{136}\right)\) |
\(\chi_{1156}(359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{136}\right)\) | \(e\left(\frac{35}{136}\right)\) | \(e\left(\frac{81}{136}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{5}{136}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{13}{136}\right)\) |
\(\chi_{1156}(383,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{136}\right)\) | \(e\left(\frac{21}{136}\right)\) | \(e\left(\frac{103}{136}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{3}{136}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{35}{136}\right)\) |
\(\chi_{1156}(427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{136}\right)\) | \(e\left(\frac{75}{136}\right)\) | \(e\left(\frac{57}{136}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{69}{136}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{125}{136}\right)\) |
\(\chi_{1156}(451,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{136}\right)\) | \(e\left(\frac{45}{136}\right)\) | \(e\left(\frac{7}{136}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{123}{136}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{75}{136}\right)\) |
\(\chi_{1156}(467,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{136}\right)\) | \(e\left(\frac{129}{136}\right)\) | \(e\left(\frac{11}{136}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{135}{136}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{79}{136}\right)\) |
\(\chi_{1156}(491,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{136}\right)\) | \(e\left(\frac{79}{136}\right)\) | \(e\left(\frac{109}{136}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{89}{136}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{41}{136}\right)\) |
\(\chi_{1156}(495,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{136}\right)\) | \(e\left(\frac{115}{136}\right)\) | \(e\left(\frac{33}{136}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{133}{136}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{101}{136}\right)\) |
\(\chi_{1156}(519,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{117}{136}\right)\) | \(e\left(\frac{69}{136}\right)\) | \(e\left(\frac{47}{136}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{107}{136}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{115}{136}\right)\) |
\(\chi_{1156}(535,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{105}{136}\right)\) | \(e\left(\frac{107}{136}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{39}{136}\right)\) |
\(\chi_{1156}(559,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{136}\right)\) | \(e\left(\frac{39}{136}\right)\) | \(e\left(\frac{133}{136}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{25}{136}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{65}{136}\right)\) |
\(\chi_{1156}(563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{136}\right)\) | \(e\left(\frac{19}{136}\right)\) | \(e\left(\frac{9}{136}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{61}{136}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{77}{136}\right)\) |
\(\chi_{1156}(587,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{136}\right)\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{87}{136}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{91}{136}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{19}{136}\right)\) |