Properties

Label 6001.30
Modulus $6001$
Conductor $6001$
Order $176$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(6001)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([44,61]))
 
pari: [g,chi] = znchar(Mod(30,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(6001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(176\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6001.dx

\(\chi_{6001}(30,\cdot)\) \(\chi_{6001}(47,\cdot)\) \(\chi_{6001}(72,\cdot)\) \(\chi_{6001}(98,\cdot)\) \(\chi_{6001}(157,\cdot)\) \(\chi_{6001}(259,\cdot)\) \(\chi_{6001}(310,\cdot)\) \(\chi_{6001}(429,\cdot)\) \(\chi_{6001}(446,\cdot)\) \(\chi_{6001}(608,\cdot)\) \(\chi_{6001}(659,\cdot)\) \(\chi_{6001}(676,\cdot)\) \(\chi_{6001}(744,\cdot)\) \(\chi_{6001}(1041,\cdot)\) \(\chi_{6001}(1050,\cdot)\) \(\chi_{6001}(1084,\cdot)\) \(\chi_{6001}(1109,\cdot)\) \(\chi_{6001}(1186,\cdot)\) \(\chi_{6001}(1203,\cdot)\) \(\chi_{6001}(1313,\cdot)\) \(\chi_{6001}(1347,\cdot)\) \(\chi_{6001}(1373,\cdot)\) \(\chi_{6001}(1458,\cdot)\) \(\chi_{6001}(1577,\cdot)\) \(\chi_{6001}(1645,\cdot)\) \(\chi_{6001}(1679,\cdot)\) \(\chi_{6001}(1687,\cdot)\) \(\chi_{6001}(1806,\cdot)\) \(\chi_{6001}(1857,\cdot)\) \(\chi_{6001}(1917,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((2825,3180)\) → \((i,e\left(\frac{61}{176}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{105}{176}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{61}{176}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{17}{88}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{6}{11}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{176})$
Fixed field: Number field defined by a degree 176 polynomial