Properties

Label 121.6
Modulus $121$
Conductor $121$
Order $110$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(121, base_ring=CyclotomicField(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([89]))
 
pari: [g,chi] = znchar(Mod(6,121))
 

Basic properties

Modulus: \(121\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
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 121.h

\(\chi_{121}(2,\cdot)\) \(\chi_{121}(6,\cdot)\) \(\chi_{121}(7,\cdot)\) \(\chi_{121}(8,\cdot)\) \(\chi_{121}(13,\cdot)\) \(\chi_{121}(17,\cdot)\) \(\chi_{121}(18,\cdot)\) \(\chi_{121}(19,\cdot)\) \(\chi_{121}(24,\cdot)\) \(\chi_{121}(28,\cdot)\) \(\chi_{121}(29,\cdot)\) \(\chi_{121}(30,\cdot)\) \(\chi_{121}(35,\cdot)\) \(\chi_{121}(39,\cdot)\) \(\chi_{121}(41,\cdot)\) \(\chi_{121}(46,\cdot)\) \(\chi_{121}(50,\cdot)\) \(\chi_{121}(51,\cdot)\) \(\chi_{121}(52,\cdot)\) \(\chi_{121}(57,\cdot)\) \(\chi_{121}(61,\cdot)\) \(\chi_{121}(62,\cdot)\) \(\chi_{121}(63,\cdot)\) \(\chi_{121}(68,\cdot)\) \(\chi_{121}(72,\cdot)\) \(\chi_{121}(73,\cdot)\) \(\chi_{121}(74,\cdot)\) \(\chi_{121}(79,\cdot)\) \(\chi_{121}(83,\cdot)\) \(\chi_{121}(84,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{89}{110}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 121 }(6, a) \) \(-1\)\(1\)\(e\left(\frac{89}{110}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{34}{55}\right)\)\(e\left(\frac{48}{55}\right)\)\(e\left(\frac{1}{110}\right)\)\(e\left(\frac{73}{110}\right)\)\(e\left(\frac{47}{110}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{9}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 121 }(6,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 121 }(6,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 121 }(6,·),\chi_{ 121 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 121 }(6,·)) \;\) at \(\; a,b = \) e.g. 1,2