Properties

Label 3549.1570
Modulus $3549$
Conductor $1183$
Order $78$
Real no
Primitive no
Minimal yes
Parity even

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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([0,26,29]))
 
pari: [g,chi] = znchar(Mod(1570,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(1183\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1183}(387,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.dw

\(\chi_{3549}(4,\cdot)\) \(\chi_{3549}(205,\cdot)\) \(\chi_{3549}(277,\cdot)\) \(\chi_{3549}(478,\cdot)\) \(\chi_{3549}(550,\cdot)\) \(\chi_{3549}(751,\cdot)\) \(\chi_{3549}(1024,\cdot)\) \(\chi_{3549}(1096,\cdot)\) \(\chi_{3549}(1297,\cdot)\) \(\chi_{3549}(1369,\cdot)\) \(\chi_{3549}(1570,\cdot)\) \(\chi_{3549}(1642,\cdot)\) \(\chi_{3549}(1843,\cdot)\) \(\chi_{3549}(1915,\cdot)\) \(\chi_{3549}(2116,\cdot)\) \(\chi_{3549}(2188,\cdot)\) \(\chi_{3549}(2461,\cdot)\) \(\chi_{3549}(2662,\cdot)\) \(\chi_{3549}(2734,\cdot)\) \(\chi_{3549}(2935,\cdot)\) \(\chi_{3549}(3007,\cdot)\) \(\chi_{3549}(3208,\cdot)\) \(\chi_{3549}(3280,\cdot)\) \(\chi_{3549}(3481,\cdot)\)

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

\((1184,1522,3382)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{29}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 3549 }(1570, a) \) \(1\)\(1\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{1}{78}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{49}{78}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{7}{78}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3549 }(1570,a) \;\) at \(\;a = \) e.g. 2