Properties

Label 4020.2083
Modulus $4020$
Conductor $1340$
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(4020, base_ring=CyclotomicField(132))
 
M = H._module
 
chi = DirichletCharacter(H, M([66,0,99,80]))
 
pari: [g,chi] = znchar(Mod(2083,4020))
 

Basic properties

Modulus: \(4020\)
Conductor: \(1340\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1340}(743,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4020.dn

\(\chi_{4020}(103,\cdot)\) \(\chi_{4020}(127,\cdot)\) \(\chi_{4020}(307,\cdot)\) \(\chi_{4020}(523,\cdot)\) \(\chi_{4020}(583,\cdot)\) \(\chi_{4020}(607,\cdot)\) \(\chi_{4020}(703,\cdot)\) \(\chi_{4020}(763,\cdot)\) \(\chi_{4020}(823,\cdot)\) \(\chi_{4020}(907,\cdot)\) \(\chi_{4020}(1003,\cdot)\) \(\chi_{4020}(1327,\cdot)\) \(\chi_{4020}(1363,\cdot)\) \(\chi_{4020}(1387,\cdot)\) \(\chi_{4020}(1423,\cdot)\) \(\chi_{4020}(1507,\cdot)\) \(\chi_{4020}(1567,\cdot)\) \(\chi_{4020}(1627,\cdot)\) \(\chi_{4020}(1663,\cdot)\) \(\chi_{4020}(1807,\cdot)\) \(\chi_{4020}(2083,\cdot)\) \(\chi_{4020}(2167,\cdot)\) \(\chi_{4020}(2227,\cdot)\) \(\chi_{4020}(2467,\cdot)\) \(\chi_{4020}(2563,\cdot)\) \(\chi_{4020}(2623,\cdot)\) \(\chi_{4020}(2803,\cdot)\) \(\chi_{4020}(2863,\cdot)\) \(\chi_{4020}(2887,\cdot)\) \(\chi_{4020}(2983,\cdot)\) ...

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

\((2011,2681,3217,1141)\) → \((-1,1,-i,e\left(\frac{20}{33}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 4020 }(2083, a) \) \(1\)\(1\)\(e\left(\frac{25}{132}\right)\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{101}{132}\right)\)\(e\left(\frac{71}{132}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{95}{132}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{4}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4020 }(2083,a) \;\) at \(\;a = \) e.g. 2