Properties

Label 8034.55
Modulus $8034$
Conductor $1339$
Order $51$
Real no
Primitive no
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(8034)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,17,31]))
 
pari: [g,chi] = znchar(Mod(55,8034))
 

Basic properties

Modulus: \(8034\)
Conductor: \(1339\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(51\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1339}(55,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.cs

\(\chi_{8034}(55,\cdot)\) \(\chi_{8034}(139,\cdot)\) \(\chi_{8034}(289,\cdot)\) \(\chi_{8034}(607,\cdot)\) \(\chi_{8034}(841,\cdot)\) \(\chi_{8034}(1231,\cdot)\) \(\chi_{8034}(2167,\cdot)\) \(\chi_{8034}(2245,\cdot)\) \(\chi_{8034}(2395,\cdot)\) \(\chi_{8034}(2401,\cdot)\) \(\chi_{8034}(2479,\cdot)\) \(\chi_{8034}(2707,\cdot)\) \(\chi_{8034}(2947,\cdot)\) \(\chi_{8034}(3025,\cdot)\) \(\chi_{8034}(3181,\cdot)\) \(\chi_{8034}(3253,\cdot)\) \(\chi_{8034}(3955,\cdot)\) \(\chi_{8034}(4033,\cdot)\) \(\chi_{8034}(4273,\cdot)\) \(\chi_{8034}(4345,\cdot)\) \(\chi_{8034}(4423,\cdot)\) \(\chi_{8034}(4969,\cdot)\) \(\chi_{8034}(5281,\cdot)\) \(\chi_{8034}(5827,\cdot)\) \(\chi_{8034}(5989,\cdot)\) \(\chi_{8034}(6763,\cdot)\) \(\chi_{8034}(6919,\cdot)\) \(\chi_{8034}(7159,\cdot)\) \(\chi_{8034}(7465,\cdot)\) \(\chi_{8034}(7783,\cdot)\) ...

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

\((5357,1237,5773)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{31}{51}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(e\left(\frac{31}{51}\right)\)\(e\left(\frac{5}{51}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{11}{51}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{47}{51}\right)\)\(e\left(\frac{11}{51}\right)\)\(e\left(\frac{31}{51}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{12}{17}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 51 polynomial