Properties

Label 1156.r
Modulus $1156$
Conductor $1156$
Order $136$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
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 \(-1\) \(1\) \(3\) \(5\) \(7\) \(9\) \(11\) \(13\) \(15\) \(19\) \(21\) \(23\)
\(\chi_{1156}(15,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{136}\right)\) \(e\left(\frac{87}{136}\right)\) \(e\left(\frac{77}{136}\right)\) \(e\left(\frac{47}{68}\right)\) \(e\left(\frac{129}{136}\right)\) \(e\left(\frac{25}{34}\right)\) \(e\left(\frac{67}{68}\right)\) \(e\left(\frac{23}{68}\right)\) \(e\left(\frac{31}{34}\right)\) \(e\left(\frac{9}{136}\right)\)
\(\chi_{1156}(19,\cdot)\) \(-1\) \(1\) \(e\left(\frac{75}{136}\right)\) \(e\left(\frac{107}{136}\right)\) \(e\left(\frac{65}{136}\right)\) \(e\left(\frac{7}{68}\right)\) \(e\left(\frac{93}{136}\right)\) \(e\left(\frac{3}{34}\right)\) \(e\left(\frac{23}{68}\right)\) \(e\left(\frac{15}{68}\right)\) \(e\left(\frac{1}{34}\right)\) \(e\left(\frac{133}{136}\right)\)
\(\chi_{1156}(43,\cdot)\) \(-1\) \(1\) \(e\left(\frac{45}{136}\right)\) \(e\left(\frac{37}{136}\right)\) \(e\left(\frac{39}{136}\right)\) \(e\left(\frac{45}{68}\right)\) \(e\left(\frac{83}{136}\right)\) \(e\left(\frac{29}{34}\right)\) \(e\left(\frac{41}{68}\right)\) \(e\left(\frac{9}{68}\right)\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{107}{136}\right)\)
\(\chi_{1156}(59,\cdot)\) \(-1\) \(1\) \(e\left(\frac{49}{136}\right)\) \(e\left(\frac{1}{136}\right)\) \(e\left(\frac{115}{136}\right)\) \(e\left(\frac{49}{68}\right)\) \(e\left(\frac{39}{136}\right)\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{25}{68}\right)\) \(e\left(\frac{37}{68}\right)\) \(e\left(\frac{7}{34}\right)\) \(e\left(\frac{47}{136}\right)\)
\(\chi_{1156}(83,\cdot)\) \(-1\) \(1\) \(e\left(\frac{127}{136}\right)\) \(e\left(\frac{47}{136}\right)\) \(e\left(\frac{101}{136}\right)\) \(e\left(\frac{59}{68}\right)\) \(e\left(\frac{65}{136}\right)\) \(e\left(\frac{1}{34}\right)\) \(e\left(\frac{19}{68}\right)\) \(e\left(\frac{39}{68}\right)\) \(e\left(\frac{23}{34}\right)\) \(e\left(\frac{33}{136}\right)\)
\(\chi_{1156}(87,\cdot)\) \(-1\) \(1\) \(e\left(\frac{131}{136}\right)\) \(e\left(\frac{11}{136}\right)\) \(e\left(\frac{41}{136}\right)\) \(e\left(\frac{63}{68}\right)\) \(e\left(\frac{21}{136}\right)\) \(e\left(\frac{27}{34}\right)\) \(e\left(\frac{3}{68}\right)\) \(e\left(\frac{67}{68}\right)\) \(e\left(\frac{9}{34}\right)\) \(e\left(\frac{109}{136}\right)\)
\(\chi_{1156}(111,\cdot)\) \(-1\) \(1\) \(e\left(\frac{133}{136}\right)\) \(e\left(\frac{61}{136}\right)\) \(e\left(\frac{79}{136}\right)\) \(e\left(\frac{65}{68}\right)\) \(e\left(\frac{67}{136}\right)\) \(e\left(\frac{23}{34}\right)\) \(e\left(\frac{29}{68}\right)\) \(e\left(\frac{13}{68}\right)\) \(e\left(\frac{19}{34}\right)\) \(e\left(\frac{11}{136}\right)\)
\(\chi_{1156}(127,\cdot)\) \(-1\) \(1\) \(e\left(\frac{97}{136}\right)\) \(e\left(\frac{113}{136}\right)\) \(e\left(\frac{75}{136}\right)\) \(e\left(\frac{29}{68}\right)\) \(e\left(\frac{55}{136}\right)\) \(e\left(\frac{27}{34}\right)\) \(e\left(\frac{37}{68}\right)\) \(e\left(\frac{33}{68}\right)\) \(e\left(\frac{9}{34}\right)\) \(e\left(\frac{7}{136}\right)\)
\(\chi_{1156}(151,\cdot)\) \(-1\) \(1\) \(e\left(\frac{71}{136}\right)\) \(e\left(\frac{7}{136}\right)\) \(e\left(\frac{125}{136}\right)\) \(e\left(\frac{3}{68}\right)\) \(e\left(\frac{1}{136}\right)\) \(e\left(\frac{11}{34}\right)\) \(e\left(\frac{39}{68}\right)\) \(e\left(\frac{55}{68}\right)\) \(e\left(\frac{15}{34}\right)\) \(e\left(\frac{57}{136}\right)\)
\(\chi_{1156}(195,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{136}\right)\) \(e\left(\frac{89}{136}\right)\) \(e\left(\frac{35}{136}\right)\) \(e\left(\frac{9}{68}\right)\) \(e\left(\frac{71}{136}\right)\) \(e\left(\frac{33}{34}\right)\) \(e\left(\frac{49}{68}\right)\) \(e\left(\frac{29}{68}\right)\) \(e\left(\frac{11}{34}\right)\) \(e\left(\frac{103}{136}\right)\)
\(\chi_{1156}(219,\cdot)\) \(-1\) \(1\) \(e\left(\frac{15}{136}\right)\) \(e\left(\frac{103}{136}\right)\) \(e\left(\frac{13}{136}\right)\) \(e\left(\frac{15}{68}\right)\) \(e\left(\frac{73}{136}\right)\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{59}{68}\right)\) \(e\left(\frac{3}{68}\right)\) \(e\left(\frac{7}{34}\right)\) \(e\left(\frac{81}{136}\right)\)
\(\chi_{1156}(223,\cdot)\) \(-1\) \(1\) \(e\left(\frac{107}{136}\right)\) \(e\left(\frac{91}{136}\right)\) \(e\left(\frac{129}{136}\right)\) \(e\left(\frac{39}{68}\right)\) \(e\left(\frac{13}{136}\right)\) \(e\left(\frac{7}{34}\right)\) \(e\left(\frac{31}{68}\right)\) \(e\left(\frac{35}{68}\right)\) \(e\left(\frac{25}{34}\right)\) \(e\left(\frac{61}{136}\right)\)
\(\chi_{1156}(247,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{136}\right)\) \(e\left(\frac{109}{136}\right)\) \(e\left(\frac{23}{136}\right)\) \(e\left(\frac{37}{68}\right)\) \(e\left(\frac{35}{136}\right)\) \(e\left(\frac{11}{34}\right)\) \(e\left(\frac{5}{68}\right)\) \(e\left(\frac{21}{68}\right)\) \(e\left(\frac{15}{34}\right)\) \(e\left(\frac{91}{136}\right)\)
\(\chi_{1156}(263,\cdot)\) \(-1\) \(1\) \(e\left(\frac{57}{136}\right)\) \(e\left(\frac{65}{136}\right)\) \(e\left(\frac{131}{136}\right)\) \(e\left(\frac{57}{68}\right)\) \(e\left(\frac{87}{136}\right)\) \(e\left(\frac{5}{34}\right)\) \(e\left(\frac{61}{68}\right)\) \(e\left(\frac{25}{68}\right)\) \(e\left(\frac{13}{34}\right)\) \(e\left(\frac{63}{136}\right)\)
\(\chi_{1156}(287,\cdot)\) \(-1\) \(1\) \(e\left(\frac{95}{136}\right)\) \(e\left(\frac{63}{136}\right)\) \(e\left(\frac{37}{136}\right)\) \(e\left(\frac{27}{68}\right)\) \(e\left(\frac{9}{136}\right)\) \(e\left(\frac{31}{34}\right)\) \(e\left(\frac{11}{68}\right)\) \(e\left(\frac{19}{68}\right)\) \(e\left(\frac{33}{34}\right)\) \(e\left(\frac{105}{136}\right)\)
\(\chi_{1156}(291,\cdot)\) \(-1\) \(1\) \(e\left(\frac{27}{136}\right)\) \(e\left(\frac{131}{136}\right)\) \(e\left(\frac{105}{136}\right)\) \(e\left(\frac{27}{68}\right)\) \(e\left(\frac{77}{136}\right)\) \(e\left(\frac{31}{34}\right)\) \(e\left(\frac{11}{68}\right)\) \(e\left(\frac{19}{68}\right)\) \(e\left(\frac{33}{34}\right)\) \(e\left(\frac{37}{136}\right)\)
\(\chi_{1156}(315,\cdot)\) \(-1\) \(1\) \(e\left(\frac{125}{136}\right)\) \(e\left(\frac{133}{136}\right)\) \(e\left(\frac{63}{136}\right)\) \(e\left(\frac{57}{68}\right)\) \(e\left(\frac{19}{136}\right)\) \(e\left(\frac{5}{34}\right)\) \(e\left(\frac{61}{68}\right)\) \(e\left(\frac{25}{68}\right)\) \(e\left(\frac{13}{34}\right)\) \(e\left(\frac{131}{136}\right)\)
\(\chi_{1156}(331,\cdot)\) \(-1\) \(1\) \(e\left(\frac{105}{136}\right)\) \(e\left(\frac{41}{136}\right)\) \(e\left(\frac{91}{136}\right)\) \(e\left(\frac{37}{68}\right)\) \(e\left(\frac{103}{136}\right)\) \(e\left(\frac{11}{34}\right)\) \(e\left(\frac{5}{68}\right)\) \(e\left(\frac{21}{68}\right)\) \(e\left(\frac{15}{34}\right)\) \(e\left(\frac{23}{136}\right)\)
\(\chi_{1156}(355,\cdot)\) \(-1\) \(1\) \(e\left(\frac{39}{136}\right)\) \(e\left(\frac{23}{136}\right)\) \(e\left(\frac{61}{136}\right)\) \(e\left(\frac{39}{68}\right)\) \(e\left(\frac{81}{136}\right)\) \(e\left(\frac{7}{34}\right)\) \(e\left(\frac{31}{68}\right)\) \(e\left(\frac{35}{68}\right)\) \(e\left(\frac{25}{34}\right)\) \(e\left(\frac{129}{136}\right)\)
\(\chi_{1156}(359,\cdot)\) \(-1\) \(1\) \(e\left(\frac{83}{136}\right)\) \(e\left(\frac{35}{136}\right)\) \(e\left(\frac{81}{136}\right)\) \(e\left(\frac{15}{68}\right)\) \(e\left(\frac{5}{136}\right)\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{59}{68}\right)\) \(e\left(\frac{3}{68}\right)\) \(e\left(\frac{7}{34}\right)\) \(e\left(\frac{13}{136}\right)\)
\(\chi_{1156}(383,\cdot)\) \(-1\) \(1\) \(e\left(\frac{77}{136}\right)\) \(e\left(\frac{21}{136}\right)\) \(e\left(\frac{103}{136}\right)\) \(e\left(\frac{9}{68}\right)\) \(e\left(\frac{3}{136}\right)\) \(e\left(\frac{33}{34}\right)\) \(e\left(\frac{49}{68}\right)\) \(e\left(\frac{29}{68}\right)\) \(e\left(\frac{11}{34}\right)\) \(e\left(\frac{35}{136}\right)\)
\(\chi_{1156}(427,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{136}\right)\) \(e\left(\frac{75}{136}\right)\) \(e\left(\frac{57}{136}\right)\) \(e\left(\frac{3}{68}\right)\) \(e\left(\frac{69}{136}\right)\) \(e\left(\frac{11}{34}\right)\) \(e\left(\frac{39}{68}\right)\) \(e\left(\frac{55}{68}\right)\) \(e\left(\frac{15}{34}\right)\) \(e\left(\frac{125}{136}\right)\)
\(\chi_{1156}(451,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{136}\right)\) \(e\left(\frac{45}{136}\right)\) \(e\left(\frac{7}{136}\right)\) \(e\left(\frac{29}{68}\right)\) \(e\left(\frac{123}{136}\right)\) \(e\left(\frac{27}{34}\right)\) \(e\left(\frac{37}{68}\right)\) \(e\left(\frac{33}{68}\right)\) \(e\left(\frac{9}{34}\right)\) \(e\left(\frac{75}{136}\right)\)
\(\chi_{1156}(467,\cdot)\) \(-1\) \(1\) \(e\left(\frac{65}{136}\right)\) \(e\left(\frac{129}{136}\right)\) \(e\left(\frac{11}{136}\right)\) \(e\left(\frac{65}{68}\right)\) \(e\left(\frac{135}{136}\right)\) \(e\left(\frac{23}{34}\right)\) \(e\left(\frac{29}{68}\right)\) \(e\left(\frac{13}{68}\right)\) \(e\left(\frac{19}{34}\right)\) \(e\left(\frac{79}{136}\right)\)
\(\chi_{1156}(491,\cdot)\) \(-1\) \(1\) \(e\left(\frac{63}{136}\right)\) \(e\left(\frac{79}{136}\right)\) \(e\left(\frac{109}{136}\right)\) \(e\left(\frac{63}{68}\right)\) \(e\left(\frac{89}{136}\right)\) \(e\left(\frac{27}{34}\right)\) \(e\left(\frac{3}{68}\right)\) \(e\left(\frac{67}{68}\right)\) \(e\left(\frac{9}{34}\right)\) \(e\left(\frac{41}{136}\right)\)
\(\chi_{1156}(495,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{136}\right)\) \(e\left(\frac{115}{136}\right)\) \(e\left(\frac{33}{136}\right)\) \(e\left(\frac{59}{68}\right)\) \(e\left(\frac{133}{136}\right)\) \(e\left(\frac{1}{34}\right)\) \(e\left(\frac{19}{68}\right)\) \(e\left(\frac{39}{68}\right)\) \(e\left(\frac{23}{34}\right)\) \(e\left(\frac{101}{136}\right)\)
\(\chi_{1156}(519,\cdot)\) \(-1\) \(1\) \(e\left(\frac{117}{136}\right)\) \(e\left(\frac{69}{136}\right)\) \(e\left(\frac{47}{136}\right)\) \(e\left(\frac{49}{68}\right)\) \(e\left(\frac{107}{136}\right)\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{25}{68}\right)\) \(e\left(\frac{37}{68}\right)\) \(e\left(\frac{7}{34}\right)\) \(e\left(\frac{115}{136}\right)\)
\(\chi_{1156}(535,\cdot)\) \(-1\) \(1\) \(e\left(\frac{113}{136}\right)\) \(e\left(\frac{105}{136}\right)\) \(e\left(\frac{107}{136}\right)\) \(e\left(\frac{45}{68}\right)\) \(e\left(\frac{15}{136}\right)\) \(e\left(\frac{29}{34}\right)\) \(e\left(\frac{41}{68}\right)\) \(e\left(\frac{9}{68}\right)\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{39}{136}\right)\)
\(\chi_{1156}(559,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{136}\right)\) \(e\left(\frac{39}{136}\right)\) \(e\left(\frac{133}{136}\right)\) \(e\left(\frac{7}{68}\right)\) \(e\left(\frac{25}{136}\right)\) \(e\left(\frac{3}{34}\right)\) \(e\left(\frac{23}{68}\right)\) \(e\left(\frac{15}{68}\right)\) \(e\left(\frac{1}{34}\right)\) \(e\left(\frac{65}{136}\right)\)
\(\chi_{1156}(563,\cdot)\) \(-1\) \(1\) \(e\left(\frac{115}{136}\right)\) \(e\left(\frac{19}{136}\right)\) \(e\left(\frac{9}{136}\right)\) \(e\left(\frac{47}{68}\right)\) \(e\left(\frac{61}{136}\right)\) \(e\left(\frac{25}{34}\right)\) \(e\left(\frac{67}{68}\right)\) \(e\left(\frac{23}{68}\right)\) \(e\left(\frac{31}{34}\right)\) \(e\left(\frac{77}{136}\right)\)
\(\chi_{1156}(587,\cdot)\) \(-1\) \(1\) \(e\left(\frac{69}{136}\right)\) \(e\left(\frac{93}{136}\right)\) \(e\left(\frac{87}{136}\right)\) \(e\left(\frac{1}{68}\right)\) \(e\left(\frac{91}{136}\right)\) \(e\left(\frac{15}{34}\right)\) \(e\left(\frac{13}{68}\right)\) \(e\left(\frac{41}{68}\right)\) \(e\left(\frac{5}{34}\right)\) \(e\left(\frac{19}{136}\right)\)