Properties

Label 1213.2
Modulus $1213$
Conductor $1213$
Order $1212$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1213, base_ring=CyclotomicField(1212))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(2,1213))
 

Basic properties

Modulus: \(1213\)
Conductor: \(1213\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1212\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1213.l

\(\chi_{1213}(2,\cdot)\) \(\chi_{1213}(5,\cdot)\) \(\chi_{1213}(6,\cdot)\) \(\chi_{1213}(14,\cdot)\) \(\chi_{1213}(20,\cdot)\) \(\chi_{1213}(23,\cdot)\) \(\chi_{1213}(24,\cdot)\) \(\chi_{1213}(26,\cdot)\) \(\chi_{1213}(32,\cdot)\) \(\chi_{1213}(38,\cdot)\) \(\chi_{1213}(41,\cdot)\) \(\chi_{1213}(45,\cdot)\) \(\chi_{1213}(50,\cdot)\) \(\chi_{1213}(51,\cdot)\) \(\chi_{1213}(54,\cdot)\) \(\chi_{1213}(55,\cdot)\) \(\chi_{1213}(56,\cdot)\) \(\chi_{1213}(59,\cdot)\) \(\chi_{1213}(60,\cdot)\) \(\chi_{1213}(62,\cdot)\) \(\chi_{1213}(65,\cdot)\) \(\chi_{1213}(66,\cdot)\) \(\chi_{1213}(67,\cdot)\) \(\chi_{1213}(68,\cdot)\) \(\chi_{1213}(69,\cdot)\) \(\chi_{1213}(72,\cdot)\) \(\chi_{1213}(78,\cdot)\) \(\chi_{1213}(88,\cdot)\) \(\chi_{1213}(89,\cdot)\) \(\chi_{1213}(103,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{1212})$
Fixed field: Number field defined by a degree 1212 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{1}{1212}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1213 }(2, a) \) \(-1\)\(1\)\(e\left(\frac{1}{1212}\right)\)\(e\left(\frac{232}{303}\right)\)\(e\left(\frac{1}{606}\right)\)\(e\left(\frac{209}{1212}\right)\)\(e\left(\frac{929}{1212}\right)\)\(e\left(\frac{64}{303}\right)\)\(e\left(\frac{1}{404}\right)\)\(e\left(\frac{161}{303}\right)\)\(e\left(\frac{35}{202}\right)\)\(e\left(\frac{331}{606}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1213 }(2,a) \;\) at \(\;a = \) e.g. 2