Properties

Label 6001.48
Modulus $6001$
Conductor $6001$
Order $352$
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([198,305]))
 
pari: [g,chi] = znchar(Mod(48,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(6001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(352\)
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.eo

\(\chi_{6001}(48,\cdot)\) \(\chi_{6001}(57,\cdot)\) \(\chi_{6001}(63,\cdot)\) \(\chi_{6001}(80,\cdot)\) \(\chi_{6001}(95,\cdot)\) \(\chi_{6001}(105,\cdot)\) \(\chi_{6001}(175,\cdot)\) \(\chi_{6001}(211,\cdot)\) \(\chi_{6001}(316,\cdot)\) \(\chi_{6001}(432,\cdot)\) \(\chi_{6001}(479,\cdot)\) \(\chi_{6001}(482,\cdot)\) \(\chi_{6001}(513,\cdot)\) \(\chi_{6001}(567,\cdot)\) \(\chi_{6001}(652,\cdot)\) \(\chi_{6001}(653,\cdot)\) \(\chi_{6001}(720,\cdot)\) \(\chi_{6001}(823,\cdot)\) \(\chi_{6001}(855,\cdot)\) \(\chi_{6001}(889,\cdot)\) \(\chi_{6001}(942,\cdot)\) \(\chi_{6001}(945,\cdot)\) \(\chi_{6001}(1112,\cdot)\) \(\chi_{6001}(1128,\cdot)\) \(\chi_{6001}(1133,\cdot)\) \(\chi_{6001}(1149,\cdot)\) \(\chi_{6001}(1184,\cdot)\) \(\chi_{6001}(1200,\cdot)\) \(\chi_{6001}(1204,\cdot)\) \(\chi_{6001}(1229,\cdot)\) ...

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

\((2825,3180)\) → \((e\left(\frac{9}{16}\right),e\left(\frac{305}{352}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{151}{352}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{195}{352}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{31}{32}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{151}{176}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{163}{176}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{352})$
Fixed field: Number field defined by a degree 352 polynomial