Properties

Label 4830.289
Modulus $4830$
Conductor $805$
Order $66$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(4830, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,33,22,42]))
 
pari: [g,chi] = znchar(Mod(289,4830))
 

Basic properties

Modulus: \(4830\)
Conductor: \(805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{805}(289,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4830.di

\(\chi_{4830}(289,\cdot)\) \(\chi_{4830}(499,\cdot)\) \(\chi_{4830}(739,\cdot)\) \(\chi_{4830}(949,\cdot)\) \(\chi_{4830}(1129,\cdot)\) \(\chi_{4830}(1159,\cdot)\) \(\chi_{4830}(1369,\cdot)\) \(\chi_{4830}(1549,\cdot)\) \(\chi_{4830}(1789,\cdot)\) \(\chi_{4830}(2419,\cdot)\) \(\chi_{4830}(2809,\cdot)\) \(\chi_{4830}(3019,\cdot)\) \(\chi_{4830}(3049,\cdot)\) \(\chi_{4830}(3229,\cdot)\) \(\chi_{4830}(3259,\cdot)\) \(\chi_{4830}(3439,\cdot)\) \(\chi_{4830}(3859,\cdot)\) \(\chi_{4830}(3889,\cdot)\) \(\chi_{4830}(4309,\cdot)\) \(\chi_{4830}(4489,\cdot)\)

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

\((3221,967,2761,1891)\) → \((1,-1,e\left(\frac{1}{3}\right),e\left(\frac{7}{11}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\(1\)\(1\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4830 }(289,a) \;\) at \(\;a = \) e.g. 2