Properties

Label 8034.137
Modulus $8034$
Conductor $4017$
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([102,187,24]))
 
pari: [g,chi] = znchar(Mod(137,8034))
 

Basic properties

Modulus: \(8034\)
Conductor: \(4017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(204\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4017}(137,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.em

\(\chi_{8034}(137,\cdot)\) \(\chi_{8034}(167,\cdot)\) \(\chi_{8034}(215,\cdot)\) \(\chi_{8034}(323,\cdot)\) \(\chi_{8034}(587,\cdot)\) \(\chi_{8034}(821,\cdot)\) \(\chi_{8034}(917,\cdot)\) \(\chi_{8034}(1163,\cdot)\) \(\chi_{8034}(1259,\cdot)\) \(\chi_{8034}(1415,\cdot)\) \(\chi_{8034}(1523,\cdot)\) \(\chi_{8034}(1553,\cdot)\) \(\chi_{8034}(1709,\cdot)\) \(\chi_{8034}(1727,\cdot)\) \(\chi_{8034}(1991,\cdot)\) \(\chi_{8034}(2021,\cdot)\) \(\chi_{8034}(2069,\cdot)\) \(\chi_{8034}(2177,\cdot)\) \(\chi_{8034}(2399,\cdot)\) \(\chi_{8034}(2771,\cdot)\) \(\chi_{8034}(2789,\cdot)\) \(\chi_{8034}(2897,\cdot)\) \(\chi_{8034}(2945,\cdot)\) \(\chi_{8034}(3053,\cdot)\) \(\chi_{8034}(3113,\cdot)\) \(\chi_{8034}(3257,\cdot)\) \(\chi_{8034}(3269,\cdot)\) \(\chi_{8034}(3413,\cdot)\) \(\chi_{8034}(3581,\cdot)\) \(\chi_{8034}(3677,\cdot)\) ...

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

\((5357,1237,5773)\) → \((-1,e\left(\frac{11}{12}\right),e\left(\frac{2}{17}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(e\left(\frac{59}{68}\right)\)\(e\left(\frac{113}{204}\right)\)\(e\left(\frac{19}{204}\right)\)\(e\left(\frac{29}{51}\right)\)\(e\left(\frac{203}{204}\right)\)\(e\left(\frac{25}{51}\right)\)\(e\left(\frac{25}{34}\right)\)\(e\left(\frac{29}{102}\right)\)\(e\left(\frac{65}{68}\right)\)\(e\left(\frac{43}{102}\right)\)
value at e.g. 2

Related number fields

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