Properties

Label 3600.41
Modulus $3600$
Conductor $1800$
Order $30$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,15,25,6]))
 
pari: [g,chi] = znchar(Mod(41,3600))
 

Basic properties

Modulus: \(3600\)
Conductor: \(1800\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1800}(941,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3600.ew

\(\chi_{3600}(41,\cdot)\) \(\chi_{3600}(281,\cdot)\) \(\chi_{3600}(761,\cdot)\) \(\chi_{3600}(1481,\cdot)\) \(\chi_{3600}(1721,\cdot)\) \(\chi_{3600}(2441,\cdot)\) \(\chi_{3600}(2921,\cdot)\) \(\chi_{3600}(3161,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((3151,901,2801,577)\) → \((1,-1,e\left(\frac{5}{6}\right),e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3600 }(41, a) \) \(-1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{29}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3600 }(41,a) \;\) at \(\;a = \) e.g. 2