Properties

Label 297.13
Modulus $297$
Conductor $297$
Order $90$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(297, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([40,9]))
 
pari: [g,chi] = znchar(Mod(13,297))
 

Basic properties

Modulus: \(297\)
Conductor: \(297\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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 297.w

\(\chi_{297}(7,\cdot)\) \(\chi_{297}(13,\cdot)\) \(\chi_{297}(40,\cdot)\) \(\chi_{297}(52,\cdot)\) \(\chi_{297}(61,\cdot)\) \(\chi_{297}(79,\cdot)\) \(\chi_{297}(85,\cdot)\) \(\chi_{297}(94,\cdot)\) \(\chi_{297}(106,\cdot)\) \(\chi_{297}(112,\cdot)\) \(\chi_{297}(139,\cdot)\) \(\chi_{297}(151,\cdot)\) \(\chi_{297}(160,\cdot)\) \(\chi_{297}(178,\cdot)\) \(\chi_{297}(184,\cdot)\) \(\chi_{297}(193,\cdot)\) \(\chi_{297}(205,\cdot)\) \(\chi_{297}(211,\cdot)\) \(\chi_{297}(238,\cdot)\) \(\chi_{297}(250,\cdot)\) \(\chi_{297}(259,\cdot)\) \(\chi_{297}(277,\cdot)\) \(\chi_{297}(283,\cdot)\) \(\chi_{297}(292,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((56,244)\) → \((e\left(\frac{4}{9}\right),e\left(\frac{1}{10}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 297 }(13, a) \) \(-1\)\(1\)\(e\left(\frac{49}{90}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{73}{90}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{59}{90}\right)\)\(e\left(\frac{16}{45}\right)\)\(e\left(\frac{8}{45}\right)\)\(e\left(\frac{17}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 297 }(13,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 297 }(13,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 297 }(13,·),\chi_{ 297 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 297 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2