Properties

Label 4410.577
Modulus $4410$
Conductor $245$
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([0,21,38]))
 
pari: [g,chi] = znchar(Mod(577,4410))
 

Basic properties

Modulus: \(4410\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(87,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4410.ek

\(\chi_{4410}(73,\cdot)\) \(\chi_{4410}(397,\cdot)\) \(\chi_{4410}(523,\cdot)\) \(\chi_{4410}(577,\cdot)\) \(\chi_{4410}(703,\cdot)\) \(\chi_{4410}(1027,\cdot)\) \(\chi_{4410}(1153,\cdot)\) \(\chi_{4410}(1333,\cdot)\) \(\chi_{4410}(1657,\cdot)\) \(\chi_{4410}(1837,\cdot)\) \(\chi_{4410}(1963,\cdot)\) \(\chi_{4410}(2287,\cdot)\) \(\chi_{4410}(2413,\cdot)\) \(\chi_{4410}(2467,\cdot)\) \(\chi_{4410}(2593,\cdot)\) \(\chi_{4410}(2917,\cdot)\) \(\chi_{4410}(3043,\cdot)\) \(\chi_{4410}(3097,\cdot)\) \(\chi_{4410}(3223,\cdot)\) \(\chi_{4410}(3673,\cdot)\) \(\chi_{4410}(3727,\cdot)\) \(\chi_{4410}(4177,\cdot)\) \(\chi_{4410}(4303,\cdot)\) \(\chi_{4410}(4357,\cdot)\)

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

\((3431,2647,1081)\) → \((1,i,e\left(\frac{19}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4410 }(577, a) \) \(1\)\(1\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{47}{84}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{79}{84}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{61}{84}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{13}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4410 }(577,a) \;\) at \(\;a = \) e.g. 2