Properties

Label 2100.439
Modulus $2100$
Conductor $700$
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(2100, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,0,9,25]))
 
pari: [g,chi] = znchar(Mod(439,2100))
 

Basic properties

Modulus: \(2100\)
Conductor: \(700\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{700}(439,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2100.cy

\(\chi_{2100}(19,\cdot)\) \(\chi_{2100}(439,\cdot)\) \(\chi_{2100}(619,\cdot)\) \(\chi_{2100}(859,\cdot)\) \(\chi_{2100}(1039,\cdot)\) \(\chi_{2100}(1279,\cdot)\) \(\chi_{2100}(1459,\cdot)\) \(\chi_{2100}(1879,\cdot)\)

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: 30.30.639471349555952501681804656982421875000000000000000000000000000000.1

Values on generators

\((1051,701,1177,1501)\) → \((-1,1,e\left(\frac{3}{10}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 2100 }(439, a) \) \(1\)\(1\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{7}{10}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2100 }(439,a) \;\) at \(\;a = \) e.g. 2