Properties

Label 304.61
Modulus $304$
Conductor $304$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(304, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,27,4]))
 
pari: [g,chi] = znchar(Mod(61,304))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 304.bi

\(\chi_{304}(5,\cdot)\) \(\chi_{304}(61,\cdot)\) \(\chi_{304}(85,\cdot)\) \(\chi_{304}(93,\cdot)\) \(\chi_{304}(101,\cdot)\) \(\chi_{304}(149,\cdot)\) \(\chi_{304}(157,\cdot)\) \(\chi_{304}(213,\cdot)\) \(\chi_{304}(237,\cdot)\) \(\chi_{304}(245,\cdot)\) \(\chi_{304}(253,\cdot)\) \(\chi_{304}(301,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.36.52733281945045886724167383478270850720626086921526306402773390818541568.1

Values on generators

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

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\(1\)\(1\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{13}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 304 }(61,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 304 }(61,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 304 }(61,·),\chi_{ 304 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 304 }(61,·)) \;\) at \(\; a,b = \) e.g. 1,2