Properties

Label 2805.71
Modulus $2805$
Conductor $561$
Order $80$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2805, base_ring=CyclotomicField(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([40,0,32,5]))
 
pari: [g,chi] = znchar(Mod(71,2805))
 

Basic properties

Modulus: \(2805\)
Conductor: \(561\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{561}(71,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2805.fd

\(\chi_{2805}(71,\cdot)\) \(\chi_{2805}(146,\cdot)\) \(\chi_{2805}(311,\cdot)\) \(\chi_{2805}(401,\cdot)\) \(\chi_{2805}(521,\cdot)\) \(\chi_{2805}(566,\cdot)\) \(\chi_{2805}(581,\cdot)\) \(\chi_{2805}(641,\cdot)\) \(\chi_{2805}(686,\cdot)\) \(\chi_{2805}(806,\cdot)\) \(\chi_{2805}(896,\cdot)\) \(\chi_{2805}(911,\cdot)\) \(\chi_{2805}(1061,\cdot)\) \(\chi_{2805}(1076,\cdot)\) \(\chi_{2805}(1136,\cdot)\) \(\chi_{2805}(1346,\cdot)\) \(\chi_{2805}(1391,\cdot)\) \(\chi_{2805}(1406,\cdot)\) \(\chi_{2805}(1571,\cdot)\) \(\chi_{2805}(1676,\cdot)\) \(\chi_{2805}(1796,\cdot)\) \(\chi_{2805}(1841,\cdot)\) \(\chi_{2805}(1901,\cdot)\) \(\chi_{2805}(1961,\cdot)\) \(\chi_{2805}(2051,\cdot)\) \(\chi_{2805}(2171,\cdot)\) \(\chi_{2805}(2216,\cdot)\) \(\chi_{2805}(2336,\cdot)\) \(\chi_{2805}(2561,\cdot)\) \(\chi_{2805}(2621,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((1871,562,1531,496)\) → \((-1,1,e\left(\frac{2}{5}\right),e\left(\frac{1}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(19\)\(23\)\(26\)
\( \chi_{ 2805 }(71, a) \) \(1\)\(1\)\(e\left(\frac{31}{40}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{39}{80}\right)\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{21}{80}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{3}{40}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{17}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2805 }(71,a) \;\) at \(\;a = \) e.g. 2