Properties

Label 8015.81
Modulus $8015$
Conductor $1603$
Order $57$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8015, base_ring=CyclotomicField(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,76,74]))
 
pari: [g,chi] = znchar(Mod(81,8015))
 

Basic properties

Modulus: \(8015\)
Conductor: \(1603\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(57\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1603}(81,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8015.ef

\(\chi_{8015}(81,\cdot)\) \(\chi_{8015}(151,\cdot)\) \(\chi_{8015}(361,\cdot)\) \(\chi_{8015}(611,\cdot)\) \(\chi_{8015}(641,\cdot)\) \(\chi_{8015}(816,\cdot)\) \(\chi_{8015}(1096,\cdot)\) \(\chi_{8015}(1271,\cdot)\) \(\chi_{8015}(1341,\cdot)\) \(\chi_{8015}(1411,\cdot)\) \(\chi_{8015}(1976,\cdot)\) \(\chi_{8015}(2081,\cdot)\) \(\chi_{8015}(2116,\cdot)\) \(\chi_{8015}(2461,\cdot)\) \(\chi_{8015}(2921,\cdot)\) \(\chi_{8015}(2991,\cdot)\) \(\chi_{8015}(3161,\cdot)\) \(\chi_{8015}(3201,\cdot)\) \(\chi_{8015}(3231,\cdot)\) \(\chi_{8015}(3831,\cdot)\) \(\chi_{8015}(3896,\cdot)\) \(\chi_{8015}(4281,\cdot)\) \(\chi_{8015}(4426,\cdot)\) \(\chi_{8015}(4631,\cdot)\) \(\chi_{8015}(4671,\cdot)\) \(\chi_{8015}(5086,\cdot)\) \(\chi_{8015}(5121,\cdot)\) \(\chi_{8015}(5196,\cdot)\) \(\chi_{8015}(5231,\cdot)\) \(\chi_{8015}(6036,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 57 polynomial

Values on generators

\((3207,4581,4586)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{37}{57}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 8015 }(81, a) \) \(1\)\(1\)\(e\left(\frac{55}{57}\right)\)\(e\left(\frac{13}{19}\right)\)\(e\left(\frac{53}{57}\right)\)\(e\left(\frac{37}{57}\right)\)\(e\left(\frac{17}{19}\right)\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{47}{57}\right)\)\(e\left(\frac{35}{57}\right)\)\(e\left(\frac{18}{19}\right)\)\(e\left(\frac{49}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8015 }(81,a) \;\) at \(\;a = \) e.g. 2