Properties

Label 6013.74
Modulus $6013$
Conductor $6013$
Order $429$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6013, base_ring=CyclotomicField(858))
 
M = H._module
 
chi = DirichletCharacter(H, M([572,642]))
 
pari: [g,chi] = znchar(Mod(74,6013))
 

Basic properties

Modulus: \(6013\)
Conductor: \(6013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(429\)
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 6013.cn

\(\chi_{6013}(74,\cdot)\) \(\chi_{6013}(79,\cdot)\) \(\chi_{6013}(121,\cdot)\) \(\chi_{6013}(149,\cdot)\) \(\chi_{6013}(163,\cdot)\) \(\chi_{6013}(219,\cdot)\) \(\chi_{6013}(228,\cdot)\) \(\chi_{6013}(235,\cdot)\) \(\chi_{6013}(249,\cdot)\) \(\chi_{6013}(270,\cdot)\) \(\chi_{6013}(291,\cdot)\) \(\chi_{6013}(303,\cdot)\) \(\chi_{6013}(340,\cdot)\) \(\chi_{6013}(347,\cdot)\) \(\chi_{6013}(354,\cdot)\) \(\chi_{6013}(366,\cdot)\) \(\chi_{6013}(387,\cdot)\) \(\chi_{6013}(401,\cdot)\) \(\chi_{6013}(417,\cdot)\) \(\chi_{6013}(466,\cdot)\) \(\chi_{6013}(471,\cdot)\) \(\chi_{6013}(480,\cdot)\) \(\chi_{6013}(494,\cdot)\) \(\chi_{6013}(499,\cdot)\) \(\chi_{6013}(508,\cdot)\) \(\chi_{6013}(522,\cdot)\) \(\chi_{6013}(527,\cdot)\) \(\chi_{6013}(571,\cdot)\) \(\chi_{6013}(585,\cdot)\) \(\chi_{6013}(604,\cdot)\) ...

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: $\Q(\zeta_{429})$
Fixed field: Number field defined by a degree 429 polynomial (not computed)

Values on generators

\((5155,3438)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{107}{143}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 6013 }(74, a) \) \(1\)\(1\)\(e\left(\frac{35}{429}\right)\)\(e\left(\frac{304}{429}\right)\)\(e\left(\frac{70}{429}\right)\)\(e\left(\frac{86}{429}\right)\)\(e\left(\frac{113}{143}\right)\)\(e\left(\frac{35}{143}\right)\)\(e\left(\frac{179}{429}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{103}{429}\right)\)\(e\left(\frac{34}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6013 }(74,a) \;\) at \(\;a = \) e.g. 2