Properties

Label 4005.16
Modulus $4005$
Conductor $801$
Order $33$
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(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,0,48]))
 
pari: [g,chi] = znchar(Mod(16,4005))
 

Basic properties

Modulus: \(4005\)
Conductor: \(801\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{801}(16,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4005.cm

\(\chi_{4005}(16,\cdot)\) \(\chi_{4005}(121,\cdot)\) \(\chi_{4005}(256,\cdot)\) \(\chi_{4005}(331,\cdot)\) \(\chi_{4005}(601,\cdot)\) \(\chi_{4005}(751,\cdot)\) \(\chi_{4005}(1291,\cdot)\) \(\chi_{4005}(1426,\cdot)\) \(\chi_{4005}(1456,\cdot)\) \(\chi_{4005}(1591,\cdot)\) \(\chi_{4005}(1606,\cdot)\) \(\chi_{4005}(1966,\cdot)\) \(\chi_{4005}(2086,\cdot)\) \(\chi_{4005}(2626,\cdot)\) \(\chi_{4005}(2686,\cdot)\) \(\chi_{4005}(2761,\cdot)\) \(\chi_{4005}(2941,\cdot)\) \(\chi_{4005}(3001,\cdot)\) \(\chi_{4005}(3271,\cdot)\) \(\chi_{4005}(3301,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 33 polynomial

Values on generators

\((3116,802,181)\) → \((e\left(\frac{2}{3}\right),1,e\left(\frac{8}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 4005 }(16, a) \) \(1\)\(1\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{5}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4005 }(16,a) \;\) at \(\;a = \) e.g. 2