Properties

Label 4005.106
Modulus $4005$
Conductor $801$
Order $132$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4005, base_ring=CyclotomicField(132))
 
M = H._module
 
chi = DirichletCharacter(H, M([88,0,9]))
 
pari: [g,chi] = znchar(Mod(106,4005))
 

Basic properties

Modulus: \(4005\)
Conductor: \(801\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{801}(106,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4005.dz

\(\chi_{4005}(106,\cdot)\) \(\chi_{4005}(196,\cdot)\) \(\chi_{4005}(346,\cdot)\) \(\chi_{4005}(376,\cdot)\) \(\chi_{4005}(436,\cdot)\) \(\chi_{4005}(466,\cdot)\) \(\chi_{4005}(481,\cdot)\) \(\chi_{4005}(691,\cdot)\) \(\chi_{4005}(781,\cdot)\) \(\chi_{4005}(796,\cdot)\) \(\chi_{4005}(841,\cdot)\) \(\chi_{4005}(961,\cdot)\) \(\chi_{4005}(1021,\cdot)\) \(\chi_{4005}(1051,\cdot)\) \(\chi_{4005}(1471,\cdot)\) \(\chi_{4005}(1651,\cdot)\) \(\chi_{4005}(1681,\cdot)\) \(\chi_{4005}(1696,\cdot)\) \(\chi_{4005}(1771,\cdot)\) \(\chi_{4005}(1816,\cdot)\) \(\chi_{4005}(2011,\cdot)\) \(\chi_{4005}(2056,\cdot)\) \(\chi_{4005}(2131,\cdot)\) \(\chi_{4005}(2146,\cdot)\) \(\chi_{4005}(2176,\cdot)\) \(\chi_{4005}(2356,\cdot)\) \(\chi_{4005}(2776,\cdot)\) \(\chi_{4005}(2806,\cdot)\) \(\chi_{4005}(2866,\cdot)\) \(\chi_{4005}(2986,\cdot)\) ...

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

Values on generators

\((3116,802,181)\) → \((e\left(\frac{2}{3}\right),1,e\left(\frac{3}{44}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 4005 }(106, a) \) \(1\)\(1\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{25}{132}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{119}{132}\right)\)\(e\left(\frac{125}{132}\right)\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{17}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4005 }(106,a) \;\) at \(\;a = \) e.g. 2