Properties

Label 115.k
Modulus $115$
Conductor $115$
Order $44$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(115, base_ring=CyclotomicField(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,4]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(2,115))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(115\)
Conductor: \(115\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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: \(\Q(\zeta_{44})\)
Fixed field: 44.0.342865339180420288801608222738062084913425127327306009945459663867950439453125.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(13\)
\(\chi_{115}(2,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{1}{44}\right)\)
\(\chi_{115}(3,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{19}{44}\right)\)
\(\chi_{115}(8,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{3}{44}\right)\)
\(\chi_{115}(12,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{21}{44}\right)\)
\(\chi_{115}(13,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{7}{44}\right)\)
\(\chi_{115}(18,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{39}{44}\right)\)
\(\chi_{115}(27,\cdot)\) \(-1\) \(1\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{13}{44}\right)\)
\(\chi_{115}(32,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{5}{44}\right)\)
\(\chi_{115}(48,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{23}{44}\right)\)
\(\chi_{115}(52,\cdot)\) \(-1\) \(1\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{9}{44}\right)\)
\(\chi_{115}(58,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{43}{44}\right)\)
\(\chi_{115}(62,\cdot)\) \(-1\) \(1\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{37}{44}\right)\)
\(\chi_{115}(72,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{41}{44}\right)\)
\(\chi_{115}(73,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{35}{44}\right)\)
\(\chi_{115}(77,\cdot)\) \(-1\) \(1\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{25}{44}\right)\)
\(\chi_{115}(78,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{27}{44}\right)\)
\(\chi_{115}(82,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{29}{44}\right)\)
\(\chi_{115}(87,\cdot)\) \(-1\) \(1\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{17}{44}\right)\)
\(\chi_{115}(98,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{31}{44}\right)\)
\(\chi_{115}(108,\cdot)\) \(-1\) \(1\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{15}{44}\right)\)