Properties

Label 2366.53
Modulus $2366$
Conductor $1183$
Order $39$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2366, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([52,60]))
 
pari: [g,chi] = znchar(Mod(53,2366))
 

Basic properties

Modulus: \(2366\)
Conductor: \(1183\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(39\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1183}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2366.bk

\(\chi_{2366}(53,\cdot)\) \(\chi_{2366}(79,\cdot)\) \(\chi_{2366}(235,\cdot)\) \(\chi_{2366}(261,\cdot)\) \(\chi_{2366}(417,\cdot)\) \(\chi_{2366}(443,\cdot)\) \(\chi_{2366}(599,\cdot)\) \(\chi_{2366}(625,\cdot)\) \(\chi_{2366}(781,\cdot)\) \(\chi_{2366}(807,\cdot)\) \(\chi_{2366}(963,\cdot)\) \(\chi_{2366}(989,\cdot)\) \(\chi_{2366}(1145,\cdot)\) \(\chi_{2366}(1171,\cdot)\) \(\chi_{2366}(1327,\cdot)\) \(\chi_{2366}(1509,\cdot)\) \(\chi_{2366}(1535,\cdot)\) \(\chi_{2366}(1717,\cdot)\) \(\chi_{2366}(1873,\cdot)\) \(\chi_{2366}(1899,\cdot)\) \(\chi_{2366}(2055,\cdot)\) \(\chi_{2366}(2081,\cdot)\) \(\chi_{2366}(2237,\cdot)\) \(\chi_{2366}(2263,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 39 polynomial

Values on generators

\((339,2199)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{10}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 2366 }(53, a) \) \(1\)\(1\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{4}{39}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{2}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2366 }(53,a) \;\) at \(\;a = \) e.g. 2