Properties

Label 4235.1299
Modulus $4235$
Conductor $4235$
Order $66$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(4235, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([33,44,36]))
 
pari: [g,chi] = znchar(Mod(1299,4235))
 

Basic properties

Modulus: \(4235\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4235.cn

\(\chi_{4235}(144,\cdot)\) \(\chi_{4235}(254,\cdot)\) \(\chi_{4235}(529,\cdot)\) \(\chi_{4235}(639,\cdot)\) \(\chi_{4235}(914,\cdot)\) \(\chi_{4235}(1024,\cdot)\) \(\chi_{4235}(1299,\cdot)\) \(\chi_{4235}(1409,\cdot)\) \(\chi_{4235}(1684,\cdot)\) \(\chi_{4235}(1794,\cdot)\) \(\chi_{4235}(2069,\cdot)\) \(\chi_{4235}(2454,\cdot)\) \(\chi_{4235}(2564,\cdot)\) \(\chi_{4235}(2839,\cdot)\) \(\chi_{4235}(2949,\cdot)\) \(\chi_{4235}(3224,\cdot)\) \(\chi_{4235}(3334,\cdot)\) \(\chi_{4235}(3609,\cdot)\) \(\chi_{4235}(3719,\cdot)\) \(\chi_{4235}(4104,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((2542,1816,2906)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{6}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{61}{66}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{59}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(1299,a) \;\) at \(\;a = \) e.g. 2