Properties

Label 4410.139
Modulus $4410$
Conductor $2205$
Order $42$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4410, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([14,21,15]))
 
pari: [g,chi] = znchar(Mod(139,4410))
 

Basic properties

Modulus: \(4410\)
Conductor: \(2205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2205}(139,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4410.ds

\(\chi_{4410}(139,\cdot)\) \(\chi_{4410}(349,\cdot)\) \(\chi_{4410}(769,\cdot)\) \(\chi_{4410}(1399,\cdot)\) \(\chi_{4410}(1609,\cdot)\) \(\chi_{4410}(2029,\cdot)\) \(\chi_{4410}(2239,\cdot)\) \(\chi_{4410}(2659,\cdot)\) \(\chi_{4410}(2869,\cdot)\) \(\chi_{4410}(3289,\cdot)\) \(\chi_{4410}(3499,\cdot)\) \(\chi_{4410}(4129,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((3431,2647,1081)\) → \((e\left(\frac{1}{3}\right),-1,e\left(\frac{5}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4410 }(139, a) \) \(-1\)\(1\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(-1\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{41}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4410 }(139,a) \;\) at \(\;a = \) e.g. 2