Properties

Label 370.53
Modulus $370$
Conductor $185$
Order $36$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(370, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([27,4]))
 
pari: [g,chi] = znchar(Mod(53,370))
 

Basic properties

Modulus: \(370\)
Conductor: \(185\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{185}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 370.z

\(\chi_{370}(7,\cdot)\) \(\chi_{370}(33,\cdot)\) \(\chi_{370}(53,\cdot)\) \(\chi_{370}(83,\cdot)\) \(\chi_{370}(107,\cdot)\) \(\chi_{370}(123,\cdot)\) \(\chi_{370}(127,\cdot)\) \(\chi_{370}(157,\cdot)\) \(\chi_{370}(197,\cdot)\) \(\chi_{370}(293,\cdot)\) \(\chi_{370}(303,\cdot)\) \(\chi_{370}(367,\cdot)\)

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: 36.0.1134084166835624937413663701523229292665702790961273014545440673828125.1

Values on generators

\((297,261)\) → \((-i,e\left(\frac{1}{9}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\(-1\)\(1\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 370 }(53,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 370 }(53,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 370 }(53,·),\chi_{ 370 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 370 }(53,·)) \;\) at \(\; a,b = \) e.g. 1,2