Properties

Label 4235.1168
Modulus $4235$
Conductor $4235$
Order $220$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4235, base_ring=CyclotomicField(220))
 
M = H._module
 
chi = DirichletCharacter(H, M([165,110,82]))
 
pari: [g,chi] = znchar(Mod(1168,4235))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4235.df

\(\chi_{4235}(13,\cdot)\) \(\chi_{4235}(62,\cdot)\) \(\chi_{4235}(83,\cdot)\) \(\chi_{4235}(167,\cdot)\) \(\chi_{4235}(237,\cdot)\) \(\chi_{4235}(272,\cdot)\) \(\chi_{4235}(293,\cdot)\) \(\chi_{4235}(398,\cdot)\) \(\chi_{4235}(447,\cdot)\) \(\chi_{4235}(468,\cdot)\) \(\chi_{4235}(503,\cdot)\) \(\chi_{4235}(552,\cdot)\) \(\chi_{4235}(622,\cdot)\) \(\chi_{4235}(657,\cdot)\) \(\chi_{4235}(678,\cdot)\) \(\chi_{4235}(783,\cdot)\) \(\chi_{4235}(832,\cdot)\) \(\chi_{4235}(853,\cdot)\) \(\chi_{4235}(888,\cdot)\) \(\chi_{4235}(937,\cdot)\) \(\chi_{4235}(1007,\cdot)\) \(\chi_{4235}(1042,\cdot)\) \(\chi_{4235}(1063,\cdot)\) \(\chi_{4235}(1168,\cdot)\) \(\chi_{4235}(1217,\cdot)\) \(\chi_{4235}(1238,\cdot)\) \(\chi_{4235}(1273,\cdot)\) \(\chi_{4235}(1392,\cdot)\) \(\chi_{4235}(1427,\cdot)\) \(\chi_{4235}(1448,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ 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

\((2542,1816,2906)\) → \((-i,-1,e\left(\frac{41}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 4235 }(1168, a) \) \(-1\)\(1\)\(e\left(\frac{27}{220}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{27}{110}\right)\)\(e\left(\frac{37}{55}\right)\)\(e\left(\frac{81}{220}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{87}{220}\right)\)\(e\left(\frac{27}{55}\right)\)\(e\left(\frac{113}{220}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(1168,a) \;\) at \(\;a = \) e.g. 2