Properties

Label 8100.127
Modulus $8100$
Conductor $2700$
Order $180$
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(180))
 
M = H._module
 
chi = DirichletCharacter(H, M([90,40,9]))
 
pari: [g,chi] = znchar(Mod(127,8100))
 

Basic properties

Modulus: \(8100\)
Conductor: \(2700\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2700}(2527,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8100.df

\(\chi_{8100}(127,\cdot)\) \(\chi_{8100}(523,\cdot)\) \(\chi_{8100}(667,\cdot)\) \(\chi_{8100}(847,\cdot)\) \(\chi_{8100}(883,\cdot)\) \(\chi_{8100}(1063,\cdot)\) \(\chi_{8100}(1387,\cdot)\) \(\chi_{8100}(1423,\cdot)\) \(\chi_{8100}(1603,\cdot)\) \(\chi_{8100}(1747,\cdot)\) \(\chi_{8100}(1927,\cdot)\) \(\chi_{8100}(1963,\cdot)\) \(\chi_{8100}(2287,\cdot)\) \(\chi_{8100}(2467,\cdot)\) \(\chi_{8100}(2503,\cdot)\) \(\chi_{8100}(2683,\cdot)\) \(\chi_{8100}(2827,\cdot)\) \(\chi_{8100}(3223,\cdot)\) \(\chi_{8100}(3367,\cdot)\) \(\chi_{8100}(3547,\cdot)\) \(\chi_{8100}(3583,\cdot)\) \(\chi_{8100}(3763,\cdot)\) \(\chi_{8100}(4087,\cdot)\) \(\chi_{8100}(4123,\cdot)\) \(\chi_{8100}(4303,\cdot)\) \(\chi_{8100}(4447,\cdot)\) \(\chi_{8100}(4627,\cdot)\) \(\chi_{8100}(4663,\cdot)\) \(\chi_{8100}(4987,\cdot)\) \(\chi_{8100}(5167,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((4051,6401,7777)\) → \((-1,e\left(\frac{2}{9}\right),e\left(\frac{1}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8100 }(127, a) \) \(1\)\(1\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{17}{90}\right)\)\(e\left(\frac{131}{180}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{89}{180}\right)\)\(e\left(\frac{29}{90}\right)\)\(e\left(\frac{31}{90}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{44}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8100 }(127,a) \;\) at \(\;a = \) e.g. 2