Properties

Label 4410.529
Modulus $4410$
Conductor $2205$
Order $42$
Real no
Primitive no
Minimal yes
Parity even

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([28,21,34]))
 
pari: [g,chi] = znchar(Mod(529,4410))
 

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 \(\chi_{2205}(529,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4410.do

\(\chi_{4410}(499,\cdot)\) \(\chi_{4410}(529,\cdot)\) \(\chi_{4410}(1129,\cdot)\) \(\chi_{4410}(1159,\cdot)\) \(\chi_{4410}(1759,\cdot)\) \(\chi_{4410}(1789,\cdot)\) \(\chi_{4410}(2389,\cdot)\) \(\chi_{4410}(3049,\cdot)\) \(\chi_{4410}(3649,\cdot)\) \(\chi_{4410}(3679,\cdot)\) \(\chi_{4410}(4279,\cdot)\) \(\chi_{4410}(4309,\cdot)\)

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

\((3431,2647,1081)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{17}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4410 }(529, a) \) \(1\)\(1\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{5}{21}\right)\)\(1\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{1}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4410 }(529,a) \;\) at \(\;a = \) e.g. 2