Properties

Label 2520.37
Modulus $2520$
Conductor $280$
Order $12$
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(2520, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,6,0,3,4]))
 
pari: [g,chi] = znchar(Mod(37,2520))
 

Basic properties

Modulus: \(2520\)
Conductor: \(280\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{280}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2520.ik

\(\chi_{2520}(37,\cdot)\) \(\chi_{2520}(613,\cdot)\) \(\chi_{2520}(1117,\cdot)\) \(\chi_{2520}(2053,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.0.2951578112000000000.1

Values on generators

\((631,1261,281,2017,1081)\) → \((1,-1,1,i,e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(-1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(i\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(1\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2520 }(37,a) \;\) at \(\;a = \) e.g. 2