Properties

Label 4410.4187
Modulus $4410$
Conductor $2205$
Order $84$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4410, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([14,21,48]))
 
pari: [g,chi] = znchar(Mod(4187,4410))
 

Basic properties

Modulus: \(4410\)
Conductor: \(2205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2205}(1982,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4410.ec

\(\chi_{4410}(113,\cdot)\) \(\chi_{4410}(407,\cdot)\) \(\chi_{4410}(533,\cdot)\) \(\chi_{4410}(617,\cdot)\) \(\chi_{4410}(743,\cdot)\) \(\chi_{4410}(1037,\cdot)\) \(\chi_{4410}(1163,\cdot)\) \(\chi_{4410}(1247,\cdot)\) \(\chi_{4410}(1793,\cdot)\) \(\chi_{4410}(1877,\cdot)\) \(\chi_{4410}(2003,\cdot)\) \(\chi_{4410}(2297,\cdot)\) \(\chi_{4410}(2423,\cdot)\) \(\chi_{4410}(2507,\cdot)\) \(\chi_{4410}(2633,\cdot)\) \(\chi_{4410}(2927,\cdot)\) \(\chi_{4410}(3053,\cdot)\) \(\chi_{4410}(3263,\cdot)\) \(\chi_{4410}(3557,\cdot)\) \(\chi_{4410}(3683,\cdot)\) \(\chi_{4410}(3767,\cdot)\) \(\chi_{4410}(3893,\cdot)\) \(\chi_{4410}(4187,\cdot)\) \(\chi_{4410}(4397,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((3431,2647,1081)\) → \((e\left(\frac{1}{6}\right),i,e\left(\frac{4}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4410 }(4187, a) \) \(1\)\(1\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{79}{84}\right)\)\(e\left(\frac{1}{28}\right)\)\(-1\)\(e\left(\frac{25}{84}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{71}{84}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4410 }(4187,a) \;\) at \(\;a = \) e.g. 2