Properties

Label 3004.2287
Modulus $3004$
Conductor $3004$
Order $150$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3004, base_ring=CyclotomicField(150))
 
M = H._module
 
chi = DirichletCharacter(H, M([75,149]))
 
pari: [g,chi] = znchar(Mod(2287,3004))
 

Basic properties

Modulus: \(3004\)
Conductor: \(3004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(150\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3004.y

\(\chi_{3004}(11,\cdot)\) \(\chi_{3004}(83,\cdot)\) \(\chi_{3004}(223,\cdot)\) \(\chi_{3004}(243,\cdot)\) \(\chi_{3004}(251,\cdot)\) \(\chi_{3004}(379,\cdot)\) \(\chi_{3004}(455,\cdot)\) \(\chi_{3004}(467,\cdot)\) \(\chi_{3004}(551,\cdot)\) \(\chi_{3004}(567,\cdot)\) \(\chi_{3004}(571,\cdot)\) \(\chi_{3004}(699,\cdot)\) \(\chi_{3004}(719,\cdot)\) \(\chi_{3004}(859,\cdot)\) \(\chi_{3004}(863,\cdot)\) \(\chi_{3004}(871,\cdot)\) \(\chi_{3004}(1031,\cdot)\) \(\chi_{3004}(1051,\cdot)\) \(\chi_{3004}(1103,\cdot)\) \(\chi_{3004}(1195,\cdot)\) \(\chi_{3004}(1203,\cdot)\) \(\chi_{3004}(1339,\cdot)\) \(\chi_{3004}(1371,\cdot)\) \(\chi_{3004}(1383,\cdot)\) \(\chi_{3004}(1395,\cdot)\) \(\chi_{3004}(1631,\cdot)\) \(\chi_{3004}(1875,\cdot)\) \(\chi_{3004}(1879,\cdot)\) \(\chi_{3004}(2059,\cdot)\) \(\chi_{3004}(2063,\cdot)\) ...

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: $\Q(\zeta_{75})$
Fixed field: Number field defined by a degree 150 polynomial (not computed)

Values on generators

\((1503,1505)\) → \((-1,e\left(\frac{149}{150}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 3004 }(2287, a) \) \(1\)\(1\)\(e\left(\frac{37}{75}\right)\)\(e\left(\frac{7}{75}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{74}{75}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{46}{75}\right)\)\(e\left(\frac{44}{75}\right)\)\(e\left(\frac{121}{150}\right)\)\(e\left(\frac{107}{150}\right)\)\(e\left(\frac{49}{75}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3004 }(2287,a) \;\) at \(\;a = \) e.g. 2