Properties

Label 1840.1699
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([22,33,22,10]))
 
pari: [g,chi] = znchar(Mod(1699,1840))
 

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.cj

\(\chi_{1840}(19,\cdot)\) \(\chi_{1840}(99,\cdot)\) \(\chi_{1840}(339,\cdot)\) \(\chi_{1840}(379,\cdot)\) \(\chi_{1840}(419,\cdot)\) \(\chi_{1840}(539,\cdot)\) \(\chi_{1840}(619,\cdot)\) \(\chi_{1840}(659,\cdot)\) \(\chi_{1840}(779,\cdot)\) \(\chi_{1840}(819,\cdot)\) \(\chi_{1840}(939,\cdot)\) \(\chi_{1840}(1019,\cdot)\) \(\chi_{1840}(1259,\cdot)\) \(\chi_{1840}(1299,\cdot)\) \(\chi_{1840}(1339,\cdot)\) \(\chi_{1840}(1459,\cdot)\) \(\chi_{1840}(1539,\cdot)\) \(\chi_{1840}(1579,\cdot)\) \(\chi_{1840}(1699,\cdot)\) \(\chi_{1840}(1739,\cdot)\)

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

\((1151,1381,737,1201)\) → \((-1,-i,-1,e\left(\frac{5}{22}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{39}{44}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{15}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1840 }(1699,a) \;\) at \(\;a = \) e.g. 2