Properties

Label 8034.397
Modulus $8034$
Conductor $1339$
Order $204$
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([0,187,182]))
 
pari: [g,chi] = znchar(Mod(397,8034))
 

Basic properties

Modulus: \(8034\)
Conductor: \(1339\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(204\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1339}(397,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.ed

\(\chi_{8034}(397,\cdot)\) \(\chi_{8034}(535,\cdot)\) \(\chi_{8034}(661,\cdot)\) \(\chi_{8034}(769,\cdot)\) \(\chi_{8034}(877,\cdot)\) \(\chi_{8034}(925,\cdot)\) \(\chi_{8034}(1207,\cdot)\) \(\chi_{8034}(1393,\cdot)\) \(\chi_{8034}(1519,\cdot)\) \(\chi_{8034}(1939,\cdot)\) \(\chi_{8034}(1969,\cdot)\) \(\chi_{8034}(2095,\cdot)\) \(\chi_{8034}(2113,\cdot)\) \(\chi_{8034}(2125,\cdot)\) \(\chi_{8034}(2203,\cdot)\) \(\chi_{8034}(2671,\cdot)\) \(\chi_{8034}(2749,\cdot)\) \(\chi_{8034}(2905,\cdot)\) \(\chi_{8034}(2983,\cdot)\) \(\chi_{8034}(3031,\cdot)\) \(\chi_{8034}(3205,\cdot)\) \(\chi_{8034}(3361,\cdot)\) \(\chi_{8034}(3439,\cdot)\) \(\chi_{8034}(3577,\cdot)\) \(\chi_{8034}(3625,\cdot)\) \(\chi_{8034}(3859,\cdot)\) \(\chi_{8034}(3889,\cdot)\) \(\chi_{8034}(3907,\cdot)\) \(\chi_{8034}(3919,\cdot)\) \(\chi_{8034}(3985,\cdot)\) ...

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

\((5357,1237,5773)\) → \((1,e\left(\frac{11}{12}\right),e\left(\frac{91}{102}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(e\left(\frac{29}{204}\right)\)\(e\left(\frac{133}{204}\right)\)\(e\left(\frac{57}{68}\right)\)\(e\left(\frac{29}{102}\right)\)\(e\left(\frac{65}{68}\right)\)\(e\left(\frac{59}{102}\right)\)\(e\left(\frac{29}{102}\right)\)\(e\left(\frac{20}{51}\right)\)\(e\left(\frac{7}{68}\right)\)\(e\left(\frac{27}{34}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{204})$
Fixed field: Number field defined by a degree 204 polynomial