Properties

Label 8034.61
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([0,34,18]))
 
pari: [g,chi] = znchar(Mod(61,8034))
 

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 \(\chi_{1339}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.cq

\(\chi_{8034}(61,\cdot)\) \(\chi_{8034}(133,\cdot)\) \(\chi_{8034}(373,\cdot)\) \(\chi_{8034}(523,\cdot)\) \(\chi_{8034}(529,\cdot)\) \(\chi_{8034}(679,\cdot)\) \(\chi_{8034}(991,\cdot)\) \(\chi_{8034}(1147,\cdot)\) \(\chi_{8034}(1465,\cdot)\) \(\chi_{8034}(1621,\cdot)\) \(\chi_{8034}(1933,\cdot)\) \(\chi_{8034}(2083,\cdot)\) \(\chi_{8034}(2239,\cdot)\) \(\chi_{8034}(2551,\cdot)\) \(\chi_{8034}(3103,\cdot)\) \(\chi_{8034}(3259,\cdot)\) \(\chi_{8034}(3721,\cdot)\) \(\chi_{8034}(3877,\cdot)\) \(\chi_{8034}(3883,\cdot)\) \(\chi_{8034}(4117,\cdot)\) \(\chi_{8034}(4501,\cdot)\) \(\chi_{8034}(4735,\cdot)\) \(\chi_{8034}(4819,\cdot)\) \(\chi_{8034}(5287,\cdot)\) \(\chi_{8034}(5365,\cdot)\) \(\chi_{8034}(5437,\cdot)\) \(\chi_{8034}(5905,\cdot)\) \(\chi_{8034}(5983,\cdot)\) \(\chi_{8034}(6067,\cdot)\) \(\chi_{8034}(6685,\cdot)\) ...

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

\((5357,1237,5773)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{6}{17}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(e\left(\frac{6}{17}\right)\)\(e\left(\frac{38}{51}\right)\)\(e\left(\frac{10}{51}\right)\)\(e\left(\frac{2}{51}\right)\)\(e\left(\frac{29}{51}\right)\)\(e\left(\frac{7}{51}\right)\)\(e\left(\frac{12}{17}\right)\)\(e\left(\frac{1}{51}\right)\)\(e\left(\frac{2}{17}\right)\)\(e\left(\frac{5}{51}\right)\)
value at e.g. 2

Related number fields

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