Properties

Label 3549.2669
Modulus $3549$
Conductor $3549$
Order $78$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([39,26,19]))
 
pari: [g,chi] = znchar(Mod(2669,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(3549\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.ds

\(\chi_{3549}(179,\cdot)\) \(\chi_{3549}(212,\cdot)\) \(\chi_{3549}(452,\cdot)\) \(\chi_{3549}(725,\cdot)\) \(\chi_{3549}(758,\cdot)\) \(\chi_{3549}(998,\cdot)\) \(\chi_{3549}(1031,\cdot)\) \(\chi_{3549}(1271,\cdot)\) \(\chi_{3549}(1304,\cdot)\) \(\chi_{3549}(1577,\cdot)\) \(\chi_{3549}(1817,\cdot)\) \(\chi_{3549}(1850,\cdot)\) \(\chi_{3549}(2090,\cdot)\) \(\chi_{3549}(2123,\cdot)\) \(\chi_{3549}(2363,\cdot)\) \(\chi_{3549}(2396,\cdot)\) \(\chi_{3549}(2636,\cdot)\) \(\chi_{3549}(2669,\cdot)\) \(\chi_{3549}(2909,\cdot)\) \(\chi_{3549}(2942,\cdot)\) \(\chi_{3549}(3182,\cdot)\) \(\chi_{3549}(3215,\cdot)\) \(\chi_{3549}(3455,\cdot)\) \(\chi_{3549}(3488,\cdot)\)

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 78 polynomial

Values on generators

\((1184,1522,3382)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{19}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 3549 }(2669, a) \) \(-1\)\(1\)\(e\left(\frac{16}{39}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{25}{39}\right)\)\(e\left(\frac{31}{78}\right)\)\(-1\)\(e\left(\frac{7}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3549 }(2669,a) \;\) at \(\;a = \) e.g. 2