Properties

Label 1840.269
Modulus $1840$
Conductor $1840$
Order $44$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,33,22,16]))
 
pari: [g,chi] = znchar(Mod(269,1840))
 

Basic properties

Modulus: \(1840\)
Conductor: \(1840\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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 1840.ck

\(\chi_{1840}(29,\cdot)\) \(\chi_{1840}(269,\cdot)\) \(\chi_{1840}(349,\cdot)\) \(\chi_{1840}(469,\cdot)\) \(\chi_{1840}(509,\cdot)\) \(\chi_{1840}(629,\cdot)\) \(\chi_{1840}(669,\cdot)\) \(\chi_{1840}(749,\cdot)\) \(\chi_{1840}(869,\cdot)\) \(\chi_{1840}(909,\cdot)\) \(\chi_{1840}(949,\cdot)\) \(\chi_{1840}(1189,\cdot)\) \(\chi_{1840}(1269,\cdot)\) \(\chi_{1840}(1389,\cdot)\) \(\chi_{1840}(1429,\cdot)\) \(\chi_{1840}(1549,\cdot)\) \(\chi_{1840}(1589,\cdot)\) \(\chi_{1840}(1669,\cdot)\) \(\chi_{1840}(1789,\cdot)\) \(\chi_{1840}(1829,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((1151,1381,737,1201)\) → \((1,-i,-1,e\left(\frac{4}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 1840 }(269, a) \) \(1\)\(1\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{35}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1840 }(269,a) \;\) at \(\;a = \) e.g. 2