Properties

Label 1840.1157
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([0,11,11,38]))
 
pari: [g,chi] = znchar(Mod(1157,1840))
 

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.da

\(\chi_{1840}(37,\cdot)\) \(\chi_{1840}(333,\cdot)\) \(\chi_{1840}(493,\cdot)\) \(\chi_{1840}(517,\cdot)\) \(\chi_{1840}(573,\cdot)\) \(\chi_{1840}(677,\cdot)\) \(\chi_{1840}(733,\cdot)\) \(\chi_{1840}(757,\cdot)\) \(\chi_{1840}(893,\cdot)\) \(\chi_{1840}(917,\cdot)\) \(\chi_{1840}(973,\cdot)\) \(\chi_{1840}(1077,\cdot)\) \(\chi_{1840}(1157,\cdot)\) \(\chi_{1840}(1213,\cdot)\) \(\chi_{1840}(1293,\cdot)\) \(\chi_{1840}(1397,\cdot)\) \(\chi_{1840}(1477,\cdot)\) \(\chi_{1840}(1533,\cdot)\) \(\chi_{1840}(1693,\cdot)\) \(\chi_{1840}(1717,\cdot)\)

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

\((1151,1381,737,1201)\) → \((1,i,i,e\left(\frac{19}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 1840 }(1157, a) \) \(1\)\(1\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{35}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1840 }(1157,a) \;\) at \(\;a = \) e.g. 2