Properties

Label 1840.77
Modulus $1840$
Conductor $1840$
Order $44$
Real no
Primitive yes
Minimal yes
Parity odd

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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([0,33,11,12]))
 
pari: [g,chi] = znchar(Mod(77,1840))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1840.ce

\(\chi_{1840}(77,\cdot)\) \(\chi_{1840}(133,\cdot)\) \(\chi_{1840}(213,\cdot)\) \(\chi_{1840}(317,\cdot)\) \(\chi_{1840}(397,\cdot)\) \(\chi_{1840}(453,\cdot)\) \(\chi_{1840}(533,\cdot)\) \(\chi_{1840}(637,\cdot)\) \(\chi_{1840}(693,\cdot)\) \(\chi_{1840}(717,\cdot)\) \(\chi_{1840}(853,\cdot)\) \(\chi_{1840}(877,\cdot)\) \(\chi_{1840}(933,\cdot)\) \(\chi_{1840}(1037,\cdot)\) \(\chi_{1840}(1093,\cdot)\) \(\chi_{1840}(1117,\cdot)\) \(\chi_{1840}(1277,\cdot)\) \(\chi_{1840}(1573,\cdot)\) \(\chi_{1840}(1733,\cdot)\) \(\chi_{1840}(1757,\cdot)\)

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

\((1151,1381,737,1201)\) → \((1,-i,i,e\left(\frac{3}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 1840 }(77, a) \) \(-1\)\(1\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{29}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1840 }(77,a) \;\) at \(\;a = \) e.g. 2