Properties

Label 85600.8129
Modulus $85600$
Conductor $2675$
Order $530$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(85600, base_ring=CyclotomicField(530))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,53,85]))
 
pari: [g,chi] = znchar(Mod(8129,85600))
 

Basic properties

Modulus: \(85600\)
Conductor: \(2675\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(530\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2675}(104,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 85600.gp

\(\chi_{85600}(129,\cdot)\) \(\chi_{85600}(609,\cdot)\) \(\chi_{85600}(769,\cdot)\) \(\chi_{85600}(929,\cdot)\) \(\chi_{85600}(1409,\cdot)\) \(\chi_{85600}(1569,\cdot)\) \(\chi_{85600}(1729,\cdot)\) \(\chi_{85600}(1889,\cdot)\) \(\chi_{85600}(2369,\cdot)\) \(\chi_{85600}(2529,\cdot)\) \(\chi_{85600}(3169,\cdot)\) \(\chi_{85600}(3489,\cdot)\) \(\chi_{85600}(4129,\cdot)\) \(\chi_{85600}(4609,\cdot)\) \(\chi_{85600}(4929,\cdot)\) \(\chi_{85600}(5089,\cdot)\) \(\chi_{85600}(5409,\cdot)\) \(\chi_{85600}(5569,\cdot)\) \(\chi_{85600}(5729,\cdot)\) \(\chi_{85600}(6529,\cdot)\) \(\chi_{85600}(6689,\cdot)\) \(\chi_{85600}(7009,\cdot)\) \(\chi_{85600}(7969,\cdot)\) \(\chi_{85600}(8129,\cdot)\) \(\chi_{85600}(8289,\cdot)\) \(\chi_{85600}(9569,\cdot)\) \(\chi_{85600}(9889,\cdot)\) \(\chi_{85600}(10369,\cdot)\) \(\chi_{85600}(10529,\cdot)\) \(\chi_{85600}(10689,\cdot)\) ...

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: $\Q(\zeta_{265})$
Fixed field: Number field defined by a degree 530 polynomial (not computed)

Values on generators

\((26751,32101,82177,16801)\) → \((1,1,e\left(\frac{1}{10}\right),e\left(\frac{17}{106}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 85600 }(8129, a) \) \(-1\)\(1\)\(e\left(\frac{491}{530}\right)\)\(e\left(\frac{21}{53}\right)\)\(e\left(\frac{226}{265}\right)\)\(e\left(\frac{34}{265}\right)\)\(e\left(\frac{77}{530}\right)\)\(e\left(\frac{252}{265}\right)\)\(e\left(\frac{82}{265}\right)\)\(e\left(\frac{171}{530}\right)\)\(e\left(\frac{23}{530}\right)\)\(e\left(\frac{413}{530}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 85600 }(8129,a) \;\) at \(\;a = \) e.g. 2