Properties

Label 1840.807
Modulus $1840$
Conductor $920$
Order $44$
Real no
Primitive no
Minimal no
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([22,22,11,4]))
 
pari: [g,chi] = znchar(Mod(807,1840))
 

Basic properties

Modulus: \(1840\)
Conductor: \(920\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{920}(347,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1840.co

\(\chi_{1840}(87,\cdot)\) \(\chi_{1840}(167,\cdot)\) \(\chi_{1840}(407,\cdot)\) \(\chi_{1840}(423,\cdot)\) \(\chi_{1840}(487,\cdot)\) \(\chi_{1840}(583,\cdot)\) \(\chi_{1840}(647,\cdot)\) \(\chi_{1840}(807,\cdot)\) \(\chi_{1840}(823,\cdot)\) \(\chi_{1840}(887,\cdot)\) \(\chi_{1840}(903,\cdot)\) \(\chi_{1840}(1047,\cdot)\) \(\chi_{1840}(1143,\cdot)\) \(\chi_{1840}(1223,\cdot)\) \(\chi_{1840}(1383,\cdot)\) \(\chi_{1840}(1527,\cdot)\) \(\chi_{1840}(1543,\cdot)\) \(\chi_{1840}(1623,\cdot)\) \(\chi_{1840}(1687,\cdot)\) \(\chi_{1840}(1783,\cdot)\)

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

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

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 1840 }(807, a) \) \(1\)\(1\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{39}{44}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{27}{44}\right)\)\(e\left(\frac{7}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1840 }(807,a) \;\) at \(\;a = \) e.g. 2