Properties

Label 950.461
Modulus $950$
Conductor $475$
Order $45$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([72,80]))
 
pari: [g,chi] = znchar(Mod(461,950))
 

Basic properties

Modulus: \(950\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{475}(461,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 950.bc

\(\chi_{950}(61,\cdot)\) \(\chi_{950}(81,\cdot)\) \(\chi_{950}(111,\cdot)\) \(\chi_{950}(131,\cdot)\) \(\chi_{950}(161,\cdot)\) \(\chi_{950}(271,\cdot)\) \(\chi_{950}(291,\cdot)\) \(\chi_{950}(321,\cdot)\) \(\chi_{950}(441,\cdot)\) \(\chi_{950}(461,\cdot)\) \(\chi_{950}(481,\cdot)\) \(\chi_{950}(491,\cdot)\) \(\chi_{950}(511,\cdot)\) \(\chi_{950}(541,\cdot)\) \(\chi_{950}(631,\cdot)\) \(\chi_{950}(671,\cdot)\) \(\chi_{950}(681,\cdot)\) \(\chi_{950}(731,\cdot)\) \(\chi_{950}(821,\cdot)\) \(\chi_{950}(841,\cdot)\) \(\chi_{950}(861,\cdot)\) \(\chi_{950}(871,\cdot)\) \(\chi_{950}(891,\cdot)\) \(\chi_{950}(921,\cdot)\)

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

Values on generators

\((77,401)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{8}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 950 }(461, a) \) \(1\)\(1\)\(e\left(\frac{7}{45}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{29}{45}\right)\)\(e\left(\frac{13}{45}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{26}{45}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{32}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 950 }(461,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 950 }(461,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 950 }(461,·),\chi_{ 950 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 950 }(461,·)) \;\) at \(\; a,b = \) e.g. 1,2