Properties

Label 8034.209
Modulus $8034$
Conductor $309$
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([17,0,13]))
 
pari: [g,chi] = znchar(Mod(209,8034))
 

Basic properties

Modulus: \(8034\)
Conductor: \(309\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(34\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{309}(209,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.cj

\(\chi_{8034}(209,\cdot)\) \(\chi_{8034}(443,\cdot)\) \(\chi_{8034}(1067,\cdot)\) \(\chi_{8034}(1223,\cdot)\) \(\chi_{8034}(2393,\cdot)\) \(\chi_{8034}(2705,\cdot)\) \(\chi_{8034}(2861,\cdot)\) \(\chi_{8034}(3797,\cdot)\) \(\chi_{8034}(3953,\cdot)\) \(\chi_{8034}(4265,\cdot)\) \(\chi_{8034}(4421,\cdot)\) \(\chi_{8034}(4811,\cdot)\) \(\chi_{8034}(6293,\cdot)\) \(\chi_{8034}(6995,\cdot)\) \(\chi_{8034}(7073,\cdot)\) \(\chi_{8034}(7541,\cdot)\)

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

\((5357,1237,5773)\) → \((-1,1,e\left(\frac{13}{34}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{9}{17}\right)\)\(e\left(\frac{14}{17}\right)\)\(e\left(\frac{9}{34}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{23}{34}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{13}{34}\right)\)\(e\left(\frac{27}{34}\right)\)\(e\left(\frac{7}{17}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{17})\)
Fixed field: 34.34.342523005011894297428856269332610453116457630461733441736562419892654124149.1