Properties

Label 3015.2981
Modulus $3015$
Conductor $603$
Order $66$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3015, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,0,32]))
 
pari: [g,chi] = znchar(Mod(2981,3015))
 

Basic properties

Modulus: \(3015\)
Conductor: \(603\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{603}(569,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3015.dc

\(\chi_{3015}(56,\cdot)\) \(\chi_{3015}(86,\cdot)\) \(\chi_{3015}(236,\cdot)\) \(\chi_{3015}(266,\cdot)\) \(\chi_{3015}(371,\cdot)\) \(\chi_{3015}(596,\cdot)\) \(\chi_{3015}(626,\cdot)\) \(\chi_{3015}(1121,\cdot)\) \(\chi_{3015}(1346,\cdot)\) \(\chi_{3015}(1361,\cdot)\) \(\chi_{3015}(1796,\cdot)\) \(\chi_{3015}(1856,\cdot)\) \(\chi_{3015}(1886,\cdot)\) \(\chi_{3015}(2036,\cdot)\) \(\chi_{3015}(2081,\cdot)\) \(\chi_{3015}(2696,\cdot)\) \(\chi_{3015}(2786,\cdot)\) \(\chi_{3015}(2831,\cdot)\) \(\chi_{3015}(2936,\cdot)\) \(\chi_{3015}(2981,\cdot)\)

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 66 polynomial

Values on generators

\((1676,1207,136)\) → \((e\left(\frac{1}{6}\right),1,e\left(\frac{16}{33}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3015 }(2981, a) \) \(-1\)\(1\)\(e\left(\frac{43}{66}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{28}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3015 }(2981,a) \;\) at \(\;a = \) e.g. 2