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Properties

Label 31.h
Modulus $31$
Conductor $31$
Order $30$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(30)) M = H._module chi = DirichletCharacter(H, M([1])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(3,31)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(31\)
Conductor: \(31\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(30\)
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: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{31}(3,\cdot)\) \(-1\) \(1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{23}{30}\right)\)
\(\chi_{31}(11,\cdot)\) \(-1\) \(1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{19}{30}\right)\)
\(\chi_{31}(12,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{17}{30}\right)\)
\(\chi_{31}(13,\cdot)\) \(-1\) \(1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{13}{30}\right)\)
\(\chi_{31}(17,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{11}{30}\right)\)
\(\chi_{31}(21,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{7}{30}\right)\)
\(\chi_{31}(22,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{1}{30}\right)\)
\(\chi_{31}(24,\cdot)\) \(-1\) \(1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{29}{30}\right)\)