Properties

Label 85600.1453
Modulus $85600$
Conductor $85600$
Order $2120$
Real no
Primitive yes
Minimal yes
Parity odd

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([0,1855,742,560]))
 
pari: [g,chi] = znchar(Mod(1453,85600))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 85600.in

\(\chi_{85600}(13,\cdot)\) \(\chi_{85600}(37,\cdot)\) \(\chi_{85600}(117,\cdot)\) \(\chi_{85600}(197,\cdot)\) \(\chi_{85600}(253,\cdot)\) \(\chi_{85600}(333,\cdot)\) \(\chi_{85600}(413,\cdot)\) \(\chi_{85600}(437,\cdot)\) \(\chi_{85600}(517,\cdot)\) \(\chi_{85600}(597,\cdot)\) \(\chi_{85600}(653,\cdot)\) \(\chi_{85600}(677,\cdot)\) \(\chi_{85600}(813,\cdot)\) \(\chi_{85600}(917,\cdot)\) \(\chi_{85600}(973,\cdot)\) \(\chi_{85600}(997,\cdot)\) \(\chi_{85600}(1053,\cdot)\) \(\chi_{85600}(1213,\cdot)\) \(\chi_{85600}(1317,\cdot)\) \(\chi_{85600}(1373,\cdot)\) \(\chi_{85600}(1453,\cdot)\) \(\chi_{85600}(1477,\cdot)\) \(\chi_{85600}(1533,\cdot)\)

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

\((26751,32101,82177,16801)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{7}{20}\right),e\left(\frac{14}{53}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 85600 }(1453, a) \) \(-1\)\(1\)\(e\left(\frac{1199}{2120}\right)\)\(e\left(\frac{91}{106}\right)\)\(e\left(\frac{139}{1060}\right)\)\(e\left(\frac{1667}{2120}\right)\)\(e\left(\frac{1003}{2120}\right)\)\(e\left(\frac{753}{1060}\right)\)\(e\left(\frac{61}{2120}\right)\)\(e\left(\frac{899}{2120}\right)\)\(e\left(\frac{253}{530}\right)\)\(e\left(\frac{1477}{2120}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 85600 }(1453,a) \;\) at \(\;a = \) e.g. 2