Properties

Label 1840.63
Modulus $1840$
Conductor $460$
Order $44$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,0,33,14]))
 
pari: [g,chi] = znchar(Mod(63,1840))
 

Basic properties

Modulus: \(1840\)
Conductor: \(460\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{460}(63,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1840.cp

\(\chi_{1840}(63,\cdot)\) \(\chi_{1840}(143,\cdot)\) \(\chi_{1840}(287,\cdot)\) \(\chi_{1840}(383,\cdot)\) \(\chi_{1840}(447,\cdot)\) \(\chi_{1840}(527,\cdot)\) \(\chi_{1840}(543,\cdot)\) \(\chi_{1840}(687,\cdot)\) \(\chi_{1840}(847,\cdot)\) \(\chi_{1840}(927,\cdot)\) \(\chi_{1840}(1023,\cdot)\) \(\chi_{1840}(1167,\cdot)\) \(\chi_{1840}(1183,\cdot)\) \(\chi_{1840}(1247,\cdot)\) \(\chi_{1840}(1263,\cdot)\) \(\chi_{1840}(1423,\cdot)\) \(\chi_{1840}(1487,\cdot)\) \(\chi_{1840}(1583,\cdot)\) \(\chi_{1840}(1647,\cdot)\) \(\chi_{1840}(1663,\cdot)\)

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: 44.0.3190796191738142789235043789002363949895144644980550209800192000000000000000000000000000000000.1

Values on generators

\((1151,1381,737,1201)\) → \((-1,1,-i,e\left(\frac{7}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 1840 }(63, a) \) \(-1\)\(1\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{5}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1840 }(63,a) \;\) at \(\;a = \) e.g. 2