Properties

Label 8034.37
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([0,119,186]))
 
pari: [g,chi] = znchar(Mod(37,8034))
 

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 \(\chi_{1339}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.ee

\(\chi_{8034}(37,\cdot)\) \(\chi_{8034}(145,\cdot)\) \(\chi_{8034}(193,\cdot)\) \(\chi_{8034}(301,\cdot)\) \(\chi_{8034}(319,\cdot)\) \(\chi_{8034}(331,\cdot)\) \(\chi_{8034}(691,\cdot)\) \(\chi_{8034}(1021,\cdot)\) \(\chi_{8034}(1033,\cdot)\) \(\chi_{8034}(1099,\cdot)\) \(\chi_{8034}(1267,\cdot)\) \(\chi_{8034}(1363,\cdot)\) \(\chi_{8034}(1567,\cdot)\) \(\chi_{8034}(1675,\cdot)\) \(\chi_{8034}(1831,\cdot)\) \(\chi_{8034}(1891,\cdot)\) \(\chi_{8034}(2047,\cdot)\) \(\chi_{8034}(2173,\cdot)\) \(\chi_{8034}(2269,\cdot)\) \(\chi_{8034}(2503,\cdot)\) \(\chi_{8034}(2767,\cdot)\) \(\chi_{8034}(2875,\cdot)\) \(\chi_{8034}(2923,\cdot)\) \(\chi_{8034}(2953,\cdot)\) \(\chi_{8034}(3127,\cdot)\) \(\chi_{8034}(3217,\cdot)\) \(\chi_{8034}(3235,\cdot)\) \(\chi_{8034}(3283,\cdot)\) \(\chi_{8034}(3391,\cdot)\) \(\chi_{8034}(3421,\cdot)\) ...

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

\((5357,1237,5773)\) → \((1,e\left(\frac{7}{12}\right),e\left(\frac{31}{34}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(e\left(\frac{11}{68}\right)\)\(e\left(\frac{13}{204}\right)\)\(e\left(\frac{143}{204}\right)\)\(e\left(\frac{101}{102}\right)\)\(e\left(\frac{175}{204}\right)\)\(e\left(\frac{73}{102}\right)\)\(e\left(\frac{11}{34}\right)\)\(e\left(\frac{38}{51}\right)\)\(e\left(\frac{15}{68}\right)\)\(e\left(\frac{23}{102}\right)\)
value at e.g. 2

Related number fields

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