↑

Properties

Label 403.cd
Modulus $403$
Conductor $403$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(403, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([15,58])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(21,403)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(403\)
Conductor: \(403\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(60\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{403}(21,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{59}{60}\right)\)
\(\chi_{403}(34,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{31}{60}\right)\)
\(\chi_{403}(44,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{41}{60}\right)\)
\(\chi_{403}(73,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{23}{60}\right)\)
\(\chi_{403}(83,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{29}{60}\right)\)
\(\chi_{403}(86,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{43}{60}\right)\)
\(\chi_{403}(96,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{1}{60}\right)\)
\(\chi_{403}(135,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{53}{60}\right)\)
\(\chi_{403}(148,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{13}{60}\right)\)
\(\chi_{403}(177,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{47}{60}\right)\)
\(\chi_{403}(203,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{7}{60}\right)\)
\(\chi_{403}(229,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{19}{60}\right)\)
\(\chi_{403}(239,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{17}{60}\right)\)
\(\chi_{403}(265,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{37}{60}\right)\)
\(\chi_{403}(291,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{49}{60}\right)\)
\(\chi_{403}(385,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{11}{60}\right)\)