Properties

Label 1840.1073
Modulus $1840$
Conductor $115$
Order $44$
Real no
Primitive no
Minimal no
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1840, base_ring=CyclotomicField(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,33,34]))
 
pari: [g,chi] = znchar(Mod(1073,1840))
 

Basic properties

Modulus: \(1840\)
Conductor: \(115\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{115}(38,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1840.ct

\(\chi_{1840}(17,\cdot)\) \(\chi_{1840}(33,\cdot)\) \(\chi_{1840}(97,\cdot)\) \(\chi_{1840}(113,\cdot)\) \(\chi_{1840}(273,\cdot)\) \(\chi_{1840}(337,\cdot)\) \(\chi_{1840}(433,\cdot)\) \(\chi_{1840}(497,\cdot)\) \(\chi_{1840}(513,\cdot)\) \(\chi_{1840}(753,\cdot)\) \(\chi_{1840}(833,\cdot)\) \(\chi_{1840}(977,\cdot)\) \(\chi_{1840}(1073,\cdot)\) \(\chi_{1840}(1137,\cdot)\) \(\chi_{1840}(1217,\cdot)\) \(\chi_{1840}(1233,\cdot)\) \(\chi_{1840}(1377,\cdot)\) \(\chi_{1840}(1537,\cdot)\) \(\chi_{1840}(1617,\cdot)\) \(\chi_{1840}(1713,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{44})\)
Fixed field: \(\Q(\zeta_{115})^+\)

Values on generators

\((1151,1381,737,1201)\) → \((1,1,-i,e\left(\frac{17}{22}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{27}{44}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{3}{44}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{9}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1840 }(1073,a) \;\) at \(\;a = \) e.g. 2