Properties

Label 1840.1803
Modulus $1840$
Conductor $1840$
Order $44$
Real no
Primitive yes
Minimal yes
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([22,11,33,20]))
 
pari: [g,chi] = znchar(Mod(1803,1840))
 

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.cy

\(\chi_{1840}(123,\cdot)\) \(\chi_{1840}(147,\cdot)\) \(\chi_{1840}(307,\cdot)\) \(\chi_{1840}(363,\cdot)\) \(\chi_{1840}(443,\cdot)\) \(\chi_{1840}(547,\cdot)\) \(\chi_{1840}(627,\cdot)\) \(\chi_{1840}(683,\cdot)\) \(\chi_{1840}(763,\cdot)\) \(\chi_{1840}(867,\cdot)\) \(\chi_{1840}(923,\cdot)\) \(\chi_{1840}(947,\cdot)\) \(\chi_{1840}(1083,\cdot)\) \(\chi_{1840}(1107,\cdot)\) \(\chi_{1840}(1163,\cdot)\) \(\chi_{1840}(1267,\cdot)\) \(\chi_{1840}(1323,\cdot)\) \(\chi_{1840}(1347,\cdot)\) \(\chi_{1840}(1507,\cdot)\) \(\chi_{1840}(1803,\cdot)\)

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: Number field defined by a degree 44 polynomial

Values on generators

\((1151,1381,737,1201)\) → \((-1,i,-i,e\left(\frac{5}{11}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{19}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1840 }(1803,a) \;\) at \(\;a = \) e.g. 2