Properties

Label 2209.1505
Modulus $2209$
Conductor $2209$
Order $47$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2209, base_ring=CyclotomicField(94))
 
M = H._module
 
chi = DirichletCharacter(H, M([30]))
 
pari: [g,chi] = znchar(Mod(1505,2209))
 

Basic properties

Modulus: \(2209\)
Conductor: \(2209\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(47\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2209.e

\(\chi_{2209}(48,\cdot)\) \(\chi_{2209}(95,\cdot)\) \(\chi_{2209}(142,\cdot)\) \(\chi_{2209}(189,\cdot)\) \(\chi_{2209}(236,\cdot)\) \(\chi_{2209}(283,\cdot)\) \(\chi_{2209}(330,\cdot)\) \(\chi_{2209}(377,\cdot)\) \(\chi_{2209}(424,\cdot)\) \(\chi_{2209}(471,\cdot)\) \(\chi_{2209}(518,\cdot)\) \(\chi_{2209}(565,\cdot)\) \(\chi_{2209}(612,\cdot)\) \(\chi_{2209}(659,\cdot)\) \(\chi_{2209}(706,\cdot)\) \(\chi_{2209}(753,\cdot)\) \(\chi_{2209}(800,\cdot)\) \(\chi_{2209}(847,\cdot)\) \(\chi_{2209}(894,\cdot)\) \(\chi_{2209}(941,\cdot)\) \(\chi_{2209}(988,\cdot)\) \(\chi_{2209}(1035,\cdot)\) \(\chi_{2209}(1082,\cdot)\) \(\chi_{2209}(1129,\cdot)\) \(\chi_{2209}(1176,\cdot)\) \(\chi_{2209}(1223,\cdot)\) \(\chi_{2209}(1270,\cdot)\) \(\chi_{2209}(1317,\cdot)\) \(\chi_{2209}(1364,\cdot)\) \(\chi_{2209}(1411,\cdot)\) ...

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: $\Q(\zeta_{47})$
Fixed field: Number field defined by a degree 47 polynomial

Values on generators

\(5\) → \(e\left(\frac{15}{47}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2209 }(1505, a) \) \(1\)\(1\)\(e\left(\frac{45}{47}\right)\)\(e\left(\frac{23}{47}\right)\)\(e\left(\frac{43}{47}\right)\)\(e\left(\frac{15}{47}\right)\)\(e\left(\frac{21}{47}\right)\)\(e\left(\frac{38}{47}\right)\)\(e\left(\frac{41}{47}\right)\)\(e\left(\frac{46}{47}\right)\)\(e\left(\frac{13}{47}\right)\)\(e\left(\frac{22}{47}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2209 }(1505,a) \;\) at \(\;a = \) e.g. 2