Properties

Label 8100.301
Modulus $8100$
Conductor $81$
Order $27$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8100, base_ring=CyclotomicField(54))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,38,0]))
 
pari: [g,chi] = znchar(Mod(301,8100))
 

Basic properties

Modulus: \(8100\)
Conductor: \(81\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(27\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{81}(58,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8100.bt

\(\chi_{8100}(301,\cdot)\) \(\chi_{8100}(601,\cdot)\) \(\chi_{8100}(1201,\cdot)\) \(\chi_{8100}(1501,\cdot)\) \(\chi_{8100}(2101,\cdot)\) \(\chi_{8100}(2401,\cdot)\) \(\chi_{8100}(3001,\cdot)\) \(\chi_{8100}(3301,\cdot)\) \(\chi_{8100}(3901,\cdot)\) \(\chi_{8100}(4201,\cdot)\) \(\chi_{8100}(4801,\cdot)\) \(\chi_{8100}(5101,\cdot)\) \(\chi_{8100}(5701,\cdot)\) \(\chi_{8100}(6001,\cdot)\) \(\chi_{8100}(6601,\cdot)\) \(\chi_{8100}(6901,\cdot)\) \(\chi_{8100}(7501,\cdot)\) \(\chi_{8100}(7801,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 27 polynomial

Values on generators

\((4051,6401,7777)\) → \((1,e\left(\frac{19}{27}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8100 }(301, a) \) \(1\)\(1\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{17}{27}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{1}{27}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{8}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8100 }(301,a) \;\) at \(\;a = \) e.g. 2