Properties

Label 304.123
Modulus $304$
Conductor $304$
Order $36$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,9,16]))
 
pari: [g,chi] = znchar(Mod(123,304))
 

Basic properties

Modulus: \(304\)
Conductor: \(304\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
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 304.bh

\(\chi_{304}(35,\cdot)\) \(\chi_{304}(43,\cdot)\) \(\chi_{304}(99,\cdot)\) \(\chi_{304}(123,\cdot)\) \(\chi_{304}(131,\cdot)\) \(\chi_{304}(139,\cdot)\) \(\chi_{304}(187,\cdot)\) \(\chi_{304}(195,\cdot)\) \(\chi_{304}(251,\cdot)\) \(\chi_{304}(275,\cdot)\) \(\chi_{304}(283,\cdot)\) \(\chi_{304}(291,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.0.52733281945045886724167383478270850720626086921526306402773390818541568.1

Values on generators

\((191,229,97)\) → \((-1,i,e\left(\frac{4}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 304 }(123, a) \) \(-1\)\(1\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{8}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 304 }(123,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 304 }(123,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 304 }(123,·),\chi_{ 304 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 304 }(123,·)) \;\) at \(\; a,b = \) e.g. 1,2