Properties

Label 8034.493
Modulus $8034$
Conductor $1339$
Order $34$
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([0,17,18]))
 
pari: [g,chi] = znchar(Mod(493,8034))
 

Basic properties

Modulus: \(8034\)
Conductor: \(1339\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(34\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1339}(493,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.co

\(\chi_{8034}(493,\cdot)\) \(\chi_{8034}(961,\cdot)\) \(\chi_{8034}(1039,\cdot)\) \(\chi_{8034}(1741,\cdot)\) \(\chi_{8034}(3223,\cdot)\) \(\chi_{8034}(3613,\cdot)\) \(\chi_{8034}(3769,\cdot)\) \(\chi_{8034}(4081,\cdot)\) \(\chi_{8034}(4237,\cdot)\) \(\chi_{8034}(5173,\cdot)\) \(\chi_{8034}(5329,\cdot)\) \(\chi_{8034}(5641,\cdot)\) \(\chi_{8034}(6811,\cdot)\) \(\chi_{8034}(6967,\cdot)\) \(\chi_{8034}(7591,\cdot)\) \(\chi_{8034}(7825,\cdot)\)

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

\((5357,1237,5773)\) → \((1,-1,e\left(\frac{9}{17}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(e\left(\frac{1}{34}\right)\)\(e\left(\frac{21}{34}\right)\)\(e\left(\frac{27}{34}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{29}{34}\right)\)\(e\left(\frac{12}{17}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{9}{17}\right)\)\(e\left(\frac{23}{34}\right)\)\(e\left(\frac{11}{17}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{17})\)
Fixed field: Number field defined by a degree 34 polynomial