Properties

Label 4025.464
Modulus $4025$
Conductor $4025$
Order $330$
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([99,110,60]))
 
pari: [g,chi] = znchar(Mod(464,4025))
 

Basic properties

Modulus: \(4025\)
Conductor: \(4025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
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.dm

\(\chi_{4025}(4,\cdot)\) \(\chi_{4025}(9,\cdot)\) \(\chi_{4025}(39,\cdot)\) \(\chi_{4025}(144,\cdot)\) \(\chi_{4025}(179,\cdot)\) \(\chi_{4025}(219,\cdot)\) \(\chi_{4025}(284,\cdot)\) \(\chi_{4025}(289,\cdot)\) \(\chi_{4025}(354,\cdot)\) \(\chi_{4025}(394,\cdot)\) \(\chi_{4025}(464,\cdot)\) \(\chi_{4025}(564,\cdot)\) \(\chi_{4025}(604,\cdot)\) \(\chi_{4025}(634,\cdot)\) \(\chi_{4025}(639,\cdot)\) \(\chi_{4025}(669,\cdot)\) \(\chi_{4025}(739,\cdot)\) \(\chi_{4025}(744,\cdot)\) \(\chi_{4025}(809,\cdot)\) \(\chi_{4025}(814,\cdot)\) \(\chi_{4025}(844,\cdot)\) \(\chi_{4025}(984,\cdot)\) \(\chi_{4025}(1089,\cdot)\) \(\chi_{4025}(1094,\cdot)\) \(\chi_{4025}(1129,\cdot)\) \(\chi_{4025}(1159,\cdot)\) \(\chi_{4025}(1269,\cdot)\) \(\chi_{4025}(1304,\cdot)\) \(\chi_{4025}(1369,\cdot)\) \(\chi_{4025}(1409,\cdot)\) ...

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 330 polynomial (not computed)

Values on generators

\((2577,1151,3501)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{1}{3}\right),e\left(\frac{2}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(464, a) \) \(1\)\(1\)\(e\left(\frac{109}{330}\right)\)\(e\left(\frac{113}{330}\right)\)\(e\left(\frac{109}{165}\right)\)\(e\left(\frac{37}{55}\right)\)\(e\left(\frac{109}{110}\right)\)\(e\left(\frac{113}{165}\right)\)\(e\left(\frac{127}{165}\right)\)\(e\left(\frac{1}{330}\right)\)\(e\left(\frac{27}{110}\right)\)\(e\left(\frac{53}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(464,a) \;\) at \(\;a = \) e.g. 2