Properties

Label 385.346
Modulus $385$
Conductor $77$
Order $30$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(385, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,5,12]))
 
pari: [g,chi] = znchar(Mod(346,385))
 

Basic properties

Modulus: \(385\)
Conductor: \(77\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{77}(38,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 385.bq

\(\chi_{385}(26,\cdot)\) \(\chi_{385}(31,\cdot)\) \(\chi_{385}(136,\cdot)\) \(\chi_{385}(201,\cdot)\) \(\chi_{385}(236,\cdot)\) \(\chi_{385}(306,\cdot)\) \(\chi_{385}(311,\cdot)\) \(\chi_{385}(346,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.0.13209167403604364499542354001933559191813355687.1

Values on generators

\((232,276,211)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 385 }(346, a) \) \(-1\)\(1\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{23}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 385 }(346,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 385 }(346,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 385 }(346,·),\chi_{ 385 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 385 }(346,·)) \;\) at \(\; a,b = \) e.g. 1,2