Properties

Label 212.97
Modulus $212$
Conductor $53$
Order $13$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(212, base_ring=CyclotomicField(26))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,4]))
 
pari: [g,chi] = znchar(Mod(97,212))
 

Basic properties

Modulus: \(212\)
Conductor: \(53\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(13\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{53}(44,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 212.g

\(\chi_{212}(13,\cdot)\) \(\chi_{212}(49,\cdot)\) \(\chi_{212}(69,\cdot)\) \(\chi_{212}(77,\cdot)\) \(\chi_{212}(81,\cdot)\) \(\chi_{212}(89,\cdot)\) \(\chi_{212}(97,\cdot)\) \(\chi_{212}(121,\cdot)\) \(\chi_{212}(153,\cdot)\) \(\chi_{212}(169,\cdot)\) \(\chi_{212}(201,\cdot)\) \(\chi_{212}(205,\cdot)\)

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: \(\Q(\zeta_{13})\)
Fixed field: Number field defined by a degree 13 polynomial

Values on generators

\((107,161)\) → \((1,e\left(\frac{2}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 212 }(97, a) \) \(1\)\(1\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{10}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 212 }(97,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 212 }(97,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 212 }(97,·),\chi_{ 212 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 212 }(97,·)) \;\) at \(\; a,b = \) e.g. 1,2