Properties

Label 4830.37
Modulus $4830$
Conductor $805$
Order $132$
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(132))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,33,44,126]))
 
pari: [g,chi] = znchar(Mod(37,4830))
 

Basic properties

Modulus: \(4830\)
Conductor: \(805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{805}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4830.dn

\(\chi_{4830}(37,\cdot)\) \(\chi_{4830}(67,\cdot)\) \(\chi_{4830}(247,\cdot)\) \(\chi_{4830}(373,\cdot)\) \(\chi_{4830}(457,\cdot)\) \(\chi_{4830}(613,\cdot)\) \(\chi_{4830}(697,\cdot)\) \(\chi_{4830}(793,\cdot)\) \(\chi_{4830}(907,\cdot)\) \(\chi_{4830}(1003,\cdot)\) \(\chi_{4830}(1033,\cdot)\) \(\chi_{4830}(1213,\cdot)\) \(\chi_{4830}(1423,\cdot)\) \(\chi_{4830}(1537,\cdot)\) \(\chi_{4830}(1663,\cdot)\) \(\chi_{4830}(1717,\cdot)\) \(\chi_{4830}(1873,\cdot)\) \(\chi_{4830}(2137,\cdot)\) \(\chi_{4830}(2167,\cdot)\) \(\chi_{4830}(2503,\cdot)\) \(\chi_{4830}(2587,\cdot)\) \(\chi_{4830}(2683,\cdot)\) \(\chi_{4830}(2767,\cdot)\) \(\chi_{4830}(2797,\cdot)\) \(\chi_{4830}(2977,\cdot)\) \(\chi_{4830}(3007,\cdot)\) \(\chi_{4830}(3103,\cdot)\) \(\chi_{4830}(3133,\cdot)\) \(\chi_{4830}(3217,\cdot)\) \(\chi_{4830}(3553,\cdot)\) ...

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: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

\((3221,967,2761,1891)\) → \((1,i,e\left(\frac{1}{3}\right),e\left(\frac{21}{22}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\(1\)\(1\)\(e\left(\frac{61}{66}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{35}{132}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{127}{132}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{11}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4830 }(37,a) \;\) at \(\;a = \) e.g. 2