Properties

Label 4410.2533
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([28,63,18]))
 
pari: [g,chi] = znchar(Mod(2533,4410))
 

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 \(\chi_{2205}(328,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4410.en

\(\chi_{4410}(13,\cdot)\) \(\chi_{4410}(223,\cdot)\) \(\chi_{4410}(517,\cdot)\) \(\chi_{4410}(643,\cdot)\) \(\chi_{4410}(727,\cdot)\) \(\chi_{4410}(853,\cdot)\) \(\chi_{4410}(1147,\cdot)\) \(\chi_{4410}(1357,\cdot)\) \(\chi_{4410}(1483,\cdot)\) \(\chi_{4410}(1777,\cdot)\) \(\chi_{4410}(1903,\cdot)\) \(\chi_{4410}(1987,\cdot)\) \(\chi_{4410}(2113,\cdot)\) \(\chi_{4410}(2407,\cdot)\) \(\chi_{4410}(2533,\cdot)\) \(\chi_{4410}(2617,\cdot)\) \(\chi_{4410}(3163,\cdot)\) \(\chi_{4410}(3247,\cdot)\) \(\chi_{4410}(3373,\cdot)\) \(\chi_{4410}(3667,\cdot)\) \(\chi_{4410}(3793,\cdot)\) \(\chi_{4410}(3877,\cdot)\) \(\chi_{4410}(4003,\cdot)\) \(\chi_{4410}(4297,\cdot)\)

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

\((3431,2647,1081)\) → \((e\left(\frac{1}{3}\right),-i,e\left(\frac{3}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4410 }(2533, a) \) \(1\)\(1\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{83}{84}\right)\)\(e\left(\frac{3}{28}\right)\)\(1\)\(e\left(\frac{5}{84}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{73}{84}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4410 }(2533,a) \;\) at \(\;a = \) e.g. 2