Properties

Label 4025.339
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([99,55,105]))
 
pari: [g,chi] = znchar(Mod(339,4025))
 

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.dl

\(\chi_{4025}(19,\cdot)\) \(\chi_{4025}(89,\cdot)\) \(\chi_{4025}(129,\cdot)\) \(\chi_{4025}(159,\cdot)\) \(\chi_{4025}(194,\cdot)\) \(\chi_{4025}(264,\cdot)\) \(\chi_{4025}(304,\cdot)\) \(\chi_{4025}(339,\cdot)\) \(\chi_{4025}(444,\cdot)\) \(\chi_{4025}(479,\cdot)\) \(\chi_{4025}(544,\cdot)\) \(\chi_{4025}(619,\cdot)\) \(\chi_{4025}(654,\cdot)\) \(\chi_{4025}(684,\cdot)\) \(\chi_{4025}(789,\cdot)\) \(\chi_{4025}(894,\cdot)\) \(\chi_{4025}(934,\cdot)\) \(\chi_{4025}(964,\cdot)\) \(\chi_{4025}(1004,\cdot)\) \(\chi_{4025}(1069,\cdot)\) \(\chi_{4025}(1109,\cdot)\) \(\chi_{4025}(1144,\cdot)\) \(\chi_{4025}(1279,\cdot)\) \(\chi_{4025}(1284,\cdot)\) \(\chi_{4025}(1354,\cdot)\) \(\chi_{4025}(1454,\cdot)\) \(\chi_{4025}(1459,\cdot)\) \(\chi_{4025}(1489,\cdot)\) \(\chi_{4025}(1529,\cdot)\) \(\chi_{4025}(1594,\cdot)\) ...

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

\((2577,1151,3501)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{1}{6}\right),e\left(\frac{7}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(339, a) \) \(1\)\(1\)\(e\left(\frac{89}{330}\right)\)\(e\left(\frac{59}{165}\right)\)\(e\left(\frac{89}{165}\right)\)\(e\left(\frac{69}{110}\right)\)\(e\left(\frac{89}{110}\right)\)\(e\left(\frac{118}{165}\right)\)\(e\left(\frac{109}{330}\right)\)\(e\left(\frac{148}{165}\right)\)\(e\left(\frac{36}{55}\right)\)\(e\left(\frac{13}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(339,a) \;\) at \(\;a = \) e.g. 2