Properties

Label 3004.49
Modulus $3004$
Conductor $751$
Order $125$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3004, base_ring=CyclotomicField(250))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,184]))
 
pari: [g,chi] = znchar(Mod(49,3004))
 

Basic properties

Modulus: \(3004\)
Conductor: \(751\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(125\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{751}(49,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3004.v

\(\chi_{3004}(45,\cdot)\) \(\chi_{3004}(49,\cdot)\) \(\chi_{3004}(93,\cdot)\) \(\chi_{3004}(125,\cdot)\) \(\chi_{3004}(165,\cdot)\) \(\chi_{3004}(185,\cdot)\) \(\chi_{3004}(189,\cdot)\) \(\chi_{3004}(237,\cdot)\) \(\chi_{3004}(249,\cdot)\) \(\chi_{3004}(305,\cdot)\) \(\chi_{3004}(341,\cdot)\) \(\chi_{3004}(429,\cdot)\) \(\chi_{3004}(433,\cdot)\) \(\chi_{3004}(445,\cdot)\) \(\chi_{3004}(457,\cdot)\) \(\chi_{3004}(493,\cdot)\) \(\chi_{3004}(525,\cdot)\) \(\chi_{3004}(545,\cdot)\) \(\chi_{3004}(605,\cdot)\) \(\chi_{3004}(617,\cdot)\) \(\chi_{3004}(681,\cdot)\) \(\chi_{3004}(693,\cdot)\) \(\chi_{3004}(729,\cdot)\) \(\chi_{3004}(745,\cdot)\) \(\chi_{3004}(761,\cdot)\) \(\chi_{3004}(777,\cdot)\) \(\chi_{3004}(789,\cdot)\) \(\chi_{3004}(793,\cdot)\) \(\chi_{3004}(797,\cdot)\) \(\chi_{3004}(845,\cdot)\) ...

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: $\Q(\zeta_{125})$
Fixed field: Number field defined by a degree 125 polynomial (not computed)

Values on generators

\((1503,1505)\) → \((1,e\left(\frac{92}{125}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 3004 }(49, a) \) \(1\)\(1\)\(e\left(\frac{92}{125}\right)\)\(e\left(\frac{87}{125}\right)\)\(e\left(\frac{17}{125}\right)\)\(e\left(\frac{59}{125}\right)\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{111}{125}\right)\)\(e\left(\frac{54}{125}\right)\)\(e\left(\frac{18}{125}\right)\)\(e\left(\frac{31}{125}\right)\)\(e\left(\frac{109}{125}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3004 }(49,a) \;\) at \(\;a = \) e.g. 2