Properties

Label 8670.353
Modulus $8670$
Conductor $4335$
Order $68$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8670, base_ring=CyclotomicField(68))
 
M = H._module
 
chi = DirichletCharacter(H, M([34,51,13]))
 
pari: [g,chi] = znchar(Mod(353,8670))
 

Basic properties

Modulus: \(8670\)
Conductor: \(4335\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(68\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4335}(353,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8670.cd

\(\chi_{8670}(353,\cdot)\) \(\chi_{8670}(497,\cdot)\) \(\chi_{8670}(863,\cdot)\) \(\chi_{8670}(1007,\cdot)\) \(\chi_{8670}(1373,\cdot)\) \(\chi_{8670}(1517,\cdot)\) \(\chi_{8670}(1883,\cdot)\) \(\chi_{8670}(2027,\cdot)\) \(\chi_{8670}(2393,\cdot)\) \(\chi_{8670}(2537,\cdot)\) \(\chi_{8670}(2903,\cdot)\) \(\chi_{8670}(3047,\cdot)\) \(\chi_{8670}(3413,\cdot)\) \(\chi_{8670}(3557,\cdot)\) \(\chi_{8670}(3923,\cdot)\) \(\chi_{8670}(4067,\cdot)\) \(\chi_{8670}(4433,\cdot)\) \(\chi_{8670}(4577,\cdot)\) \(\chi_{8670}(4943,\cdot)\) \(\chi_{8670}(5087,\cdot)\) \(\chi_{8670}(5597,\cdot)\) \(\chi_{8670}(5963,\cdot)\) \(\chi_{8670}(6473,\cdot)\) \(\chi_{8670}(6617,\cdot)\) \(\chi_{8670}(6983,\cdot)\) \(\chi_{8670}(7127,\cdot)\) \(\chi_{8670}(7493,\cdot)\) \(\chi_{8670}(7637,\cdot)\) \(\chi_{8670}(8003,\cdot)\) \(\chi_{8670}(8147,\cdot)\) ...

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: $\Q(\zeta_{68})$
Fixed field: Number field defined by a degree 68 polynomial

Values on generators

\((2891,6937,6361)\) → \((-1,-i,e\left(\frac{13}{68}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 8670 }(353, a) \) \(1\)\(1\)\(e\left(\frac{13}{34}\right)\)\(e\left(\frac{61}{68}\right)\)\(e\left(\frac{49}{68}\right)\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{13}{34}\right)\)\(e\left(\frac{61}{68}\right)\)\(e\left(\frac{49}{68}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{13}{68}\right)\)\(e\left(\frac{31}{68}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8670 }(353,a) \;\) at \(\;a = \) e.g. 2