Properties

Label 8100.71
Modulus $8100$
Conductor $2700$
Order $90$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8100, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([45,85,54]))
 
pari: [g,chi] = znchar(Mod(71,8100))
 

Basic properties

Modulus: \(8100\)
Conductor: \(2700\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2700}(2471,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8100.cs

\(\chi_{8100}(71,\cdot)\) \(\chi_{8100}(611,\cdot)\) \(\chi_{8100}(791,\cdot)\) \(\chi_{8100}(1331,\cdot)\) \(\chi_{8100}(1691,\cdot)\) \(\chi_{8100}(1871,\cdot)\) \(\chi_{8100}(2231,\cdot)\) \(\chi_{8100}(2411,\cdot)\) \(\chi_{8100}(2771,\cdot)\) \(\chi_{8100}(3311,\cdot)\) \(\chi_{8100}(3491,\cdot)\) \(\chi_{8100}(4031,\cdot)\) \(\chi_{8100}(4391,\cdot)\) \(\chi_{8100}(4571,\cdot)\) \(\chi_{8100}(4931,\cdot)\) \(\chi_{8100}(5111,\cdot)\) \(\chi_{8100}(5471,\cdot)\) \(\chi_{8100}(6011,\cdot)\) \(\chi_{8100}(6191,\cdot)\) \(\chi_{8100}(6731,\cdot)\) \(\chi_{8100}(7091,\cdot)\) \(\chi_{8100}(7271,\cdot)\) \(\chi_{8100}(7631,\cdot)\) \(\chi_{8100}(7811,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((4051,6401,7777)\) → \((-1,e\left(\frac{17}{18}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8100 }(71, a) \) \(1\)\(1\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{17}{45}\right)\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{13}{90}\right)\)\(e\left(\frac{17}{90}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{41}{90}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8100 }(71,a) \;\) at \(\;a = \) e.g. 2