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Properties

Label 1309.cg
Modulus $1309$
Conductor $187$
Order $40$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1309, base_ring=CyclotomicField(40)) M = H._module chi = DirichletCharacter(H, M([0,8,15])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(15,1309)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1309\)
Conductor: \(187\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(40\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from 187.r
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{40})\)
Fixed field: 40.40.24562817038400928776197921239227357886542077974183334844678041435576602047153.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(8\) \(9\) \(10\) \(12\) \(13\)
\(\chi_{1309}(15,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{1309}(36,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{1309}(246,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{1309}(372,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{1309}(400,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{1309}(603,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{1309}(610,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{1309}(631,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{1309}(729,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{1309}(757,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{1309}(841,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{1309}(960,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{1309}(988,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{1309}(995,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{1309}(1114,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{1309}(1226,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{9}{10}\right)\)