Properties

Label 4025.718
Modulus $4025$
Conductor $805$
Order $132$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4025, base_ring=CyclotomicField(132))
 
M = H._module
 
chi = DirichletCharacter(H, M([99,88,6]))
 
pari: [g,chi] = znchar(Mod(718,4025))
 

Basic properties

Modulus: \(4025\)
Conductor: \(805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{805}(718,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4025.cy

\(\chi_{4025}(107,\cdot)\) \(\chi_{4025}(268,\cdot)\) \(\chi_{4025}(382,\cdot)\) \(\chi_{4025}(457,\cdot)\) \(\chi_{4025}(543,\cdot)\) \(\chi_{4025}(557,\cdot)\) \(\chi_{4025}(618,\cdot)\) \(\chi_{4025}(632,\cdot)\) \(\chi_{4025}(718,\cdot)\) \(\chi_{4025}(732,\cdot)\) \(\chi_{4025}(793,\cdot)\) \(\chi_{4025}(893,\cdot)\) \(\chi_{4025}(907,\cdot)\) \(\chi_{4025}(1068,\cdot)\) \(\chi_{4025}(1157,\cdot)\) \(\chi_{4025}(1257,\cdot)\) \(\chi_{4025}(1318,\cdot)\) \(\chi_{4025}(1332,\cdot)\) \(\chi_{4025}(1418,\cdot)\) \(\chi_{4025}(1493,\cdot)\) \(\chi_{4025}(1607,\cdot)\) \(\chi_{4025}(1768,\cdot)\) \(\chi_{4025}(1782,\cdot)\) \(\chi_{4025}(1857,\cdot)\) \(\chi_{4025}(1943,\cdot)\) \(\chi_{4025}(2018,\cdot)\) \(\chi_{4025}(2307,\cdot)\) \(\chi_{4025}(2468,\cdot)\) \(\chi_{4025}(2482,\cdot)\) \(\chi_{4025}(2643,\cdot)\) ...

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: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(718, a) \) \(1\)\(1\)\(e\left(\frac{23}{132}\right)\)\(e\left(\frac{85}{132}\right)\)\(e\left(\frac{23}{66}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{5}{66}\right)\)\(e\left(\frac{131}{132}\right)\)\(e\left(\frac{39}{44}\right)\)\(e\left(\frac{23}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(718,a) \;\) at \(\;a = \) e.g. 2