Properties

Label 4235.377
Modulus $4235$
Conductor $4235$
Order $220$
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(220))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([55,110,16]))
 
pari: [g,chi] = znchar(Mod(377,4235))
 

Basic properties

Modulus: \(4235\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(220\)
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.dd

\(\chi_{4235}(48,\cdot)\) \(\chi_{4235}(97,\cdot)\) \(\chi_{4235}(223,\cdot)\) \(\chi_{4235}(258,\cdot)\) \(\chi_{4235}(328,\cdot)\) \(\chi_{4235}(377,\cdot)\) \(\chi_{4235}(412,\cdot)\) \(\chi_{4235}(433,\cdot)\) \(\chi_{4235}(482,\cdot)\) \(\chi_{4235}(587,\cdot)\) \(\chi_{4235}(643,\cdot)\) \(\chi_{4235}(713,\cdot)\) \(\chi_{4235}(762,\cdot)\) \(\chi_{4235}(797,\cdot)\) \(\chi_{4235}(818,\cdot)\) \(\chi_{4235}(867,\cdot)\) \(\chi_{4235}(972,\cdot)\) \(\chi_{4235}(993,\cdot)\) \(\chi_{4235}(1028,\cdot)\) \(\chi_{4235}(1147,\cdot)\) \(\chi_{4235}(1182,\cdot)\) \(\chi_{4235}(1203,\cdot)\) \(\chi_{4235}(1252,\cdot)\) \(\chi_{4235}(1357,\cdot)\) \(\chi_{4235}(1378,\cdot)\) \(\chi_{4235}(1413,\cdot)\) \(\chi_{4235}(1483,\cdot)\) \(\chi_{4235}(1532,\cdot)\) \(\chi_{4235}(1567,\cdot)\) \(\chi_{4235}(1588,\cdot)\) ...

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: $\Q(\zeta_{220})$
Fixed field: Number field defined by a degree 220 polynomial (not computed)

Values on generators

\((2542,1816,2906)\) → \((i,-1,e\left(\frac{4}{55}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{71}{220}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{71}{110}\right)\)\(e\left(\frac{107}{110}\right)\)\(e\left(\frac{213}{220}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{131}{220}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{69}{220}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(377,a) \;\) at \(\;a = \) e.g. 2