Properties

Label 4560.2981
Modulus $4560$
Conductor $912$
Order $36$
Real no
Primitive no
Minimal yes
Parity odd

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([0,9,18,0,20]))
 
pari: [g,chi] = znchar(Mod(2981,4560))
 

Basic properties

Modulus: \(4560\)
Conductor: \(912\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{912}(245,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4560.ja

\(\chi_{4560}(101,\cdot)\) \(\chi_{4560}(461,\cdot)\) \(\chi_{4560}(701,\cdot)\) \(\chi_{4560}(821,\cdot)\) \(\chi_{4560}(1061,\cdot)\) \(\chi_{4560}(1301,\cdot)\) \(\chi_{4560}(2381,\cdot)\) \(\chi_{4560}(2741,\cdot)\) \(\chi_{4560}(2981,\cdot)\) \(\chi_{4560}(3101,\cdot)\) \(\chi_{4560}(3341,\cdot)\) \(\chi_{4560}(3581,\cdot)\)

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: 36.0.20429953877724548562115535825002213860630960981294226252926298027107484541386752.1

Values on generators

\((1711,1141,3041,2737,1921)\) → \((1,i,-1,1,e\left(\frac{5}{9}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(-1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{1}{3}\right)\)\(i\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4560 }(2981,a) \;\) at \(\;a = \) e.g. 2