Properties

Label 1001.159
Modulus $1001$
Conductor $1001$
Order $30$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1001, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([25,12,10]))
 
pari: [g,chi] = znchar(Mod(159,1001))
 

Basic properties

Modulus: \(1001\)
Conductor: \(1001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
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 1001.cy

\(\chi_{1001}(159,\cdot)\) \(\chi_{1001}(269,\cdot)\) \(\chi_{1001}(432,\cdot)\) \(\chi_{1001}(542,\cdot)\) \(\chi_{1001}(614,\cdot)\) \(\chi_{1001}(724,\cdot)\) \(\chi_{1001}(796,\cdot)\) \(\chi_{1001}(906,\cdot)\)

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

\((430,365,925)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{2}{5}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(15\)
\( \chi_{ 1001 }(159, a) \) \(-1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{2}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1001 }(159,a) \;\) at \(\;a = \) e.g. 2