Properties

Label 2169.10
Modulus $2169$
Conductor $241$
Order $30$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2169, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,11]))
 
pari: [g,chi] = znchar(Mod(10,2169))
 

Basic properties

Modulus: \(2169\)
Conductor: \(241\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{241}(10,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2169.cc

\(\chi_{2169}(10,\cdot)\) \(\chi_{2169}(217,\cdot)\) \(\chi_{2169}(388,\cdot)\) \(\chi_{2169}(604,\cdot)\) \(\chi_{2169}(910,\cdot)\) \(\chi_{2169}(1045,\cdot)\) \(\chi_{2169}(1504,\cdot)\) \(\chi_{2169}(1828,\cdot)\)

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

\((965,730)\) → \((1,e\left(\frac{11}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 2169 }(10, a) \) \(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{11}{30}\right)\)\(1\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2169 }(10,a) \;\) at \(\;a = \) e.g. 2