Properties

Label 4025.641
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([66,220,75]))
 
pari: [g,chi] = znchar(Mod(641,4025))
 

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.dh

\(\chi_{4025}(11,\cdot)\) \(\chi_{4025}(86,\cdot)\) \(\chi_{4025}(191,\cdot)\) \(\chi_{4025}(221,\cdot)\) \(\chi_{4025}(291,\cdot)\) \(\chi_{4025}(296,\cdot)\) \(\chi_{4025}(366,\cdot)\) \(\chi_{4025}(396,\cdot)\) \(\chi_{4025}(431,\cdot)\) \(\chi_{4025}(471,\cdot)\) \(\chi_{4025}(536,\cdot)\) \(\chi_{4025}(571,\cdot)\) \(\chi_{4025}(641,\cdot)\) \(\chi_{4025}(681,\cdot)\) \(\chi_{4025}(711,\cdot)\) \(\chi_{4025}(746,\cdot)\) \(\chi_{4025}(816,\cdot)\) \(\chi_{4025}(856,\cdot)\) \(\chi_{4025}(891,\cdot)\) \(\chi_{4025}(996,\cdot)\) \(\chi_{4025}(1031,\cdot)\) \(\chi_{4025}(1096,\cdot)\) \(\chi_{4025}(1171,\cdot)\) \(\chi_{4025}(1206,\cdot)\) \(\chi_{4025}(1236,\cdot)\) \(\chi_{4025}(1341,\cdot)\) \(\chi_{4025}(1446,\cdot)\) \(\chi_{4025}(1486,\cdot)\) \(\chi_{4025}(1516,\cdot)\) \(\chi_{4025}(1556,\cdot)\) ...

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

\((2577,1151,3501)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{2}{3}\right),e\left(\frac{5}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(641, a) \) \(-1\)\(1\)\(e\left(\frac{163}{165}\right)\)\(e\left(\frac{116}{165}\right)\)\(e\left(\frac{161}{165}\right)\)\(e\left(\frac{38}{55}\right)\)\(e\left(\frac{53}{55}\right)\)\(e\left(\frac{67}{165}\right)\)\(e\left(\frac{301}{330}\right)\)\(e\left(\frac{112}{165}\right)\)\(e\left(\frac{54}{55}\right)\)\(e\left(\frac{157}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(641,a) \;\) at \(\;a = \) e.g. 2