Properties

Label 370.39
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([18,1]))
 
pari: [g,chi] = znchar(Mod(39,370))
 

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 \(\chi_{185}(39,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 370.bb

\(\chi_{370}(19,\cdot)\) \(\chi_{370}(39,\cdot)\) \(\chi_{370}(59,\cdot)\) \(\chi_{370}(69,\cdot)\) \(\chi_{370}(79,\cdot)\) \(\chi_{370}(89,\cdot)\) \(\chi_{370}(109,\cdot)\) \(\chi_{370}(129,\cdot)\) \(\chi_{370}(209,\cdot)\) \(\chi_{370}(239,\cdot)\) \(\chi_{370}(279,\cdot)\) \(\chi_{370}(309,\cdot)\)

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: 36.0.29411719834995153896864925426307140281034671856927417346954345703125.1

Values on generators

\((297,261)\) → \((-1,e\left(\frac{1}{36}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\(-1\)\(1\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 370 }(39,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 370 }(39,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 370 }(39,·),\chi_{ 370 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 370 }(39,·)) \;\) at \(\; a,b = \) e.g. 1,2