Properties

Label 4025.359
Modulus $4025$
Conductor $4025$
Order $330$
Real no
Primitive yes
Minimal yes
Parity odd

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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([231,110,315]))
 
pari: [g,chi] = znchar(Mod(359,4025))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4025.dn

\(\chi_{4025}(44,\cdot)\) \(\chi_{4025}(79,\cdot)\) \(\chi_{4025}(109,\cdot)\) \(\chi_{4025}(214,\cdot)\) \(\chi_{4025}(319,\cdot)\) \(\chi_{4025}(359,\cdot)\) \(\chi_{4025}(389,\cdot)\) \(\chi_{4025}(429,\cdot)\) \(\chi_{4025}(494,\cdot)\) \(\chi_{4025}(534,\cdot)\) \(\chi_{4025}(569,\cdot)\) \(\chi_{4025}(704,\cdot)\) \(\chi_{4025}(709,\cdot)\) \(\chi_{4025}(779,\cdot)\) \(\chi_{4025}(879,\cdot)\) \(\chi_{4025}(884,\cdot)\) \(\chi_{4025}(914,\cdot)\) \(\chi_{4025}(954,\cdot)\) \(\chi_{4025}(1019,\cdot)\) \(\chi_{4025}(1054,\cdot)\) \(\chi_{4025}(1164,\cdot)\) \(\chi_{4025}(1194,\cdot)\) \(\chi_{4025}(1229,\cdot)\) \(\chi_{4025}(1234,\cdot)\) \(\chi_{4025}(1339,\cdot)\) \(\chi_{4025}(1479,\cdot)\) \(\chi_{4025}(1509,\cdot)\) \(\chi_{4025}(1514,\cdot)\) \(\chi_{4025}(1579,\cdot)\) \(\chi_{4025}(1584,\cdot)\) ...

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

\((2577,1151,3501)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{1}{3}\right),e\left(\frac{21}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(359, a) \) \(-1\)\(1\)\(e\left(\frac{91}{330}\right)\)\(e\left(\frac{167}{330}\right)\)\(e\left(\frac{91}{165}\right)\)\(e\left(\frac{43}{55}\right)\)\(e\left(\frac{91}{110}\right)\)\(e\left(\frac{2}{165}\right)\)\(e\left(\frac{41}{330}\right)\)\(e\left(\frac{19}{330}\right)\)\(e\left(\frac{73}{110}\right)\)\(e\left(\frac{17}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(359,a) \;\) at \(\;a = \) e.g. 2