Properties

Label 4025.1271
Modulus $4025$
Conductor $4025$
Order $165$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4025, base_ring=CyclotomicField(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([198,220,270]))
 
pari: [g,chi] = znchar(Mod(1271,4025))
 

Basic properties

Modulus: \(4025\)
Conductor: \(4025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(165\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4025.dc

\(\chi_{4025}(16,\cdot)\) \(\chi_{4025}(81,\cdot)\) \(\chi_{4025}(121,\cdot)\) \(\chi_{4025}(156,\cdot)\) \(\chi_{4025}(186,\cdot)\) \(\chi_{4025}(256,\cdot)\) \(\chi_{4025}(261,\cdot)\) \(\chi_{4025}(331,\cdot)\) \(\chi_{4025}(361,\cdot)\) \(\chi_{4025}(466,\cdot)\) \(\chi_{4025}(541,\cdot)\) \(\chi_{4025}(606,\cdot)\) \(\chi_{4025}(611,\cdot)\) \(\chi_{4025}(646,\cdot)\) \(\chi_{4025}(716,\cdot)\) \(\chi_{4025}(786,\cdot)\) \(\chi_{4025}(821,\cdot)\) \(\chi_{4025}(886,\cdot)\) \(\chi_{4025}(956,\cdot)\) \(\chi_{4025}(961,\cdot)\) \(\chi_{4025}(991,\cdot)\) \(\chi_{4025}(1061,\cdot)\) \(\chi_{4025}(1066,\cdot)\) \(\chi_{4025}(1131,\cdot)\) \(\chi_{4025}(1136,\cdot)\) \(\chi_{4025}(1166,\cdot)\) \(\chi_{4025}(1271,\cdot)\) \(\chi_{4025}(1306,\cdot)\) \(\chi_{4025}(1346,\cdot)\) \(\chi_{4025}(1411,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 165 polynomial (not computed)

Values on generators

\((2577,1151,3501)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{2}{3}\right),e\left(\frac{9}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(1271, a) \) \(1\)\(1\)\(e\left(\frac{94}{165}\right)\)\(e\left(\frac{158}{165}\right)\)\(e\left(\frac{23}{165}\right)\)\(e\left(\frac{29}{55}\right)\)\(e\left(\frac{39}{55}\right)\)\(e\left(\frac{151}{165}\right)\)\(e\left(\frac{104}{165}\right)\)\(e\left(\frac{16}{165}\right)\)\(e\left(\frac{47}{55}\right)\)\(e\left(\frac{46}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(1271,a) \;\) at \(\;a = \) e.g. 2