Properties

Label 8034.31
Modulus $8034$
Conductor $1339$
Order $68$
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([0,51,38]))
 
pari: [g,chi] = znchar(Mod(31,8034))
 

Basic properties

Modulus: \(8034\)
Conductor: \(1339\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(68\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1339}(31,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.cx

\(\chi_{8034}(31,\cdot)\) \(\chi_{8034}(73,\cdot)\) \(\chi_{8034}(655,\cdot)\) \(\chi_{8034}(811,\cdot)\) \(\chi_{8034}(1555,\cdot)\) \(\chi_{8034}(1981,\cdot)\) \(\chi_{8034}(2257,\cdot)\) \(\chi_{8034}(2293,\cdot)\) \(\chi_{8034}(2335,\cdot)\) \(\chi_{8034}(2449,\cdot)\) \(\chi_{8034}(2803,\cdot)\) \(\chi_{8034}(3385,\cdot)\) \(\chi_{8034}(3505,\cdot)\) \(\chi_{8034}(3541,\cdot)\) \(\chi_{8034}(3739,\cdot)\) \(\chi_{8034}(3853,\cdot)\) \(\chi_{8034}(4009,\cdot)\) \(\chi_{8034}(4363,\cdot)\) \(\chi_{8034}(4399,\cdot)\) \(\chi_{8034}(4519,\cdot)\) \(\chi_{8034}(5689,\cdot)\) \(\chi_{8034}(5881,\cdot)\) \(\chi_{8034}(6001,\cdot)\) \(\chi_{8034}(6157,\cdot)\) \(\chi_{8034}(6583,\cdot)\) \(\chi_{8034}(6661,\cdot)\) \(\chi_{8034}(7093,\cdot)\) \(\chi_{8034}(7129,\cdot)\) \(\chi_{8034}(7249,\cdot)\) \(\chi_{8034}(7561,\cdot)\) ...

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

\((5357,1237,5773)\) → \((1,-i,e\left(\frac{19}{34}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(e\left(\frac{21}{68}\right)\)\(e\left(\frac{33}{68}\right)\)\(e\left(\frac{23}{68}\right)\)\(e\left(\frac{21}{34}\right)\)\(e\left(\frac{31}{68}\right)\)\(e\left(\frac{31}{34}\right)\)\(e\left(\frac{21}{34}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{41}{68}\right)\)\(e\left(\frac{27}{34}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{68})$
Fixed field: Number field defined by a degree 68 polynomial