Properties

Label 4410.473
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([70,63,8]))
 
pari: [g,chi] = znchar(Mod(473,4410))
 

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 \(\chi_{2205}(473,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4410.ep

\(\chi_{4410}(317,\cdot)\) \(\chi_{4410}(347,\cdot)\) \(\chi_{4410}(443,\cdot)\) \(\chi_{4410}(473,\cdot)\) \(\chi_{4410}(947,\cdot)\) \(\chi_{4410}(977,\cdot)\) \(\chi_{4410}(1073,\cdot)\) \(\chi_{4410}(1103,\cdot)\) \(\chi_{4410}(1577,\cdot)\) \(\chi_{4410}(1607,\cdot)\) \(\chi_{4410}(1703,\cdot)\) \(\chi_{4410}(2207,\cdot)\) \(\chi_{4410}(2237,\cdot)\) \(\chi_{4410}(2363,\cdot)\) \(\chi_{4410}(2837,\cdot)\) \(\chi_{4410}(2867,\cdot)\) \(\chi_{4410}(2963,\cdot)\) \(\chi_{4410}(2993,\cdot)\) \(\chi_{4410}(3467,\cdot)\) \(\chi_{4410}(3593,\cdot)\) \(\chi_{4410}(3623,\cdot)\) \(\chi_{4410}(4127,\cdot)\) \(\chi_{4410}(4223,\cdot)\) \(\chi_{4410}(4253,\cdot)\)

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

\((3431,2647,1081)\) → \((e\left(\frac{5}{6}\right),-i,e\left(\frac{2}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4410 }(473, a) \) \(1\)\(1\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{5}{84}\right)\)\(e\left(\frac{53}{84}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{67}{84}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{13}{84}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4410 }(473,a) \;\) at \(\;a = \) e.g. 2