Properties

Label 245.32
Modulus $245$
Conductor $245$
Order $84$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,8]))
 
pari: [g,chi] = znchar(Mod(32,245))
 

Basic properties

Modulus: \(245\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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 245.w

\(\chi_{245}(2,\cdot)\) \(\chi_{245}(23,\cdot)\) \(\chi_{245}(32,\cdot)\) \(\chi_{245}(37,\cdot)\) \(\chi_{245}(53,\cdot)\) \(\chi_{245}(58,\cdot)\) \(\chi_{245}(72,\cdot)\) \(\chi_{245}(88,\cdot)\) \(\chi_{245}(93,\cdot)\) \(\chi_{245}(102,\cdot)\) \(\chi_{245}(107,\cdot)\) \(\chi_{245}(123,\cdot)\) \(\chi_{245}(137,\cdot)\) \(\chi_{245}(142,\cdot)\) \(\chi_{245}(158,\cdot)\) \(\chi_{245}(163,\cdot)\) \(\chi_{245}(172,\cdot)\) \(\chi_{245}(193,\cdot)\) \(\chi_{245}(198,\cdot)\) \(\chi_{245}(207,\cdot)\) \(\chi_{245}(212,\cdot)\) \(\chi_{245}(228,\cdot)\) \(\chi_{245}(233,\cdot)\) \(\chi_{245}(242,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((197,101)\) → \((i,e\left(\frac{2}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 245 }(32, a) \) \(-1\)\(1\)\(e\left(\frac{61}{84}\right)\)\(e\left(\frac{71}{84}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{25}{84}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{19}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 245 }(32,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 245 }(32,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 245 }(32,·),\chi_{ 245 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 245 }(32,·)) \;\) at \(\; a,b = \) e.g. 1,2