Properties

Label 4560.4297
Modulus $4560$
Conductor $760$
Order $36$
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(4560, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,18,0,9,26]))
 
pari: [g,chi] = znchar(Mod(4297,4560))
 

Basic properties

Modulus: \(4560\)
Conductor: \(760\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{760}(117,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4560.ix

\(\chi_{4560}(553,\cdot)\) \(\chi_{4560}(697,\cdot)\) \(\chi_{4560}(793,\cdot)\) \(\chi_{4560}(2233,\cdot)\) \(\chi_{4560}(2377,\cdot)\) \(\chi_{4560}(2473,\cdot)\) \(\chi_{4560}(2617,\cdot)\) \(\chi_{4560}(2713,\cdot)\) \(\chi_{4560}(3433,\cdot)\) \(\chi_{4560}(4057,\cdot)\) \(\chi_{4560}(4297,\cdot)\) \(\chi_{4560}(4537,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.36.4031181156993454136731178943694064571490658196389888000000000000000000000000000.1

Values on generators

\((1711,1141,3041,2737,1921)\) → \((1,-1,1,i,e\left(\frac{13}{18}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{5}{6}\right)\)\(i\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{29}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4560 }(4297,a) \;\) at \(\;a = \) e.g. 2