Properties

Label 85600.379
Modulus $85600$
Conductor $85600$
Order $2120$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(85600, base_ring=CyclotomicField(2120))
 
M = H._module
 
chi = DirichletCharacter(H, M([1060,265,212,660]))
 
pari: [g,chi] = znchar(Mod(379,85600))
 

Basic properties

Modulus: \(85600\)
Conductor: \(85600\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2120\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 85600.ig

\(\chi_{85600}(59,\cdot)\) \(\chi_{85600}(139,\cdot)\) \(\chi_{85600}(179,\cdot)\) \(\chi_{85600}(219,\cdot)\) \(\chi_{85600}(259,\cdot)\) \(\chi_{85600}(339,\cdot)\) \(\chi_{85600}(379,\cdot)\) \(\chi_{85600}(419,\cdot)\) \(\chi_{85600}(459,\cdot)\) \(\chi_{85600}(619,\cdot)\) \(\chi_{85600}(659,\cdot)\) \(\chi_{85600}(739,\cdot)\) \(\chi_{85600}(819,\cdot)\) \(\chi_{85600}(1059,\cdot)\) \(\chi_{85600}(1179,\cdot)\) \(\chi_{85600}(1259,\cdot)\) \(\chi_{85600}(1339,\cdot)\) \(\chi_{85600}(1379,\cdot)\) \(\chi_{85600}(1419,\cdot)\) \(\chi_{85600}(1459,\cdot)\)

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: $\Q(\zeta_{2120})$
Fixed field: Number field defined by a degree 2120 polynomial (not computed)

Values on generators

\((26751,32101,82177,16801)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{1}{10}\right),e\left(\frac{33}{106}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 85600 }(379, a) \) \(1\)\(1\)\(e\left(\frac{779}{2120}\right)\)\(e\left(\frac{135}{212}\right)\)\(e\left(\frac{779}{1060}\right)\)\(e\left(\frac{1217}{2120}\right)\)\(e\left(\frac{283}{2120}\right)\)\(e\left(\frac{439}{530}\right)\)\(e\left(\frac{971}{2120}\right)\)\(e\left(\frac{9}{2120}\right)\)\(e\left(\frac{691}{1060}\right)\)\(e\left(\frac{217}{2120}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 85600 }(379,a) \;\) at \(\;a = \) e.g. 2