Properties

Label 3234.365
Modulus $3234$
Conductor $1617$
Order $70$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3234, base_ring=CyclotomicField(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,40,7]))
 
pari: [g,chi] = znchar(Mod(365,3234))
 

Basic properties

Modulus: \(3234\)
Conductor: \(1617\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1617}(365,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3234.by

\(\chi_{3234}(29,\cdot)\) \(\chi_{3234}(239,\cdot)\) \(\chi_{3234}(281,\cdot)\) \(\chi_{3234}(365,\cdot)\) \(\chi_{3234}(701,\cdot)\) \(\chi_{3234}(743,\cdot)\) \(\chi_{3234}(827,\cdot)\) \(\chi_{3234}(953,\cdot)\) \(\chi_{3234}(1163,\cdot)\) \(\chi_{3234}(1205,\cdot)\) \(\chi_{3234}(1289,\cdot)\) \(\chi_{3234}(1415,\cdot)\) \(\chi_{3234}(1625,\cdot)\) \(\chi_{3234}(1751,\cdot)\) \(\chi_{3234}(1877,\cdot)\) \(\chi_{3234}(2087,\cdot)\) \(\chi_{3234}(2129,\cdot)\) \(\chi_{3234}(2213,\cdot)\) \(\chi_{3234}(2339,\cdot)\) \(\chi_{3234}(2591,\cdot)\) \(\chi_{3234}(2675,\cdot)\) \(\chi_{3234}(2801,\cdot)\) \(\chi_{3234}(3011,\cdot)\) \(\chi_{3234}(3053,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((1079,199,2059)\) → \((-1,e\left(\frac{4}{7}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3234 }(365, a) \) \(1\)\(1\)\(e\left(\frac{33}{70}\right)\)\(e\left(\frac{67}{70}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{13}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3234 }(365,a) \;\) at \(\;a = \) e.g. 2