Properties

Label 4025.2557
Modulus $4025$
Conductor $805$
Order $132$
Real no
Primitive no
Minimal no
Parity odd

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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([33,44,24]))
 
pari: [g,chi] = znchar(Mod(2557,4025))
 

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 \(\chi_{805}(142,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4025.cz

\(\chi_{4025}(18,\cdot)\) \(\chi_{4025}(32,\cdot)\) \(\chi_{4025}(193,\cdot)\) \(\chi_{4025}(282,\cdot)\) \(\chi_{4025}(443,\cdot)\) \(\chi_{4025}(807,\cdot)\) \(\chi_{4025}(968,\cdot)\) \(\chi_{4025}(982,\cdot)\) \(\chi_{4025}(1143,\cdot)\) \(\chi_{4025}(1432,\cdot)\) \(\chi_{4025}(1507,\cdot)\) \(\chi_{4025}(1593,\cdot)\) \(\chi_{4025}(1668,\cdot)\) \(\chi_{4025}(1682,\cdot)\) \(\chi_{4025}(1843,\cdot)\) \(\chi_{4025}(1957,\cdot)\) \(\chi_{4025}(2032,\cdot)\) \(\chi_{4025}(2118,\cdot)\) \(\chi_{4025}(2132,\cdot)\) \(\chi_{4025}(2193,\cdot)\) \(\chi_{4025}(2293,\cdot)\) \(\chi_{4025}(2382,\cdot)\) \(\chi_{4025}(2543,\cdot)\) \(\chi_{4025}(2557,\cdot)\) \(\chi_{4025}(2657,\cdot)\) \(\chi_{4025}(2718,\cdot)\) \(\chi_{4025}(2732,\cdot)\) \(\chi_{4025}(2818,\cdot)\) \(\chi_{4025}(2832,\cdot)\) \(\chi_{4025}(2893,\cdot)\) ...

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

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

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(2557, a) \) \(-1\)\(1\)\(e\left(\frac{37}{132}\right)\)\(e\left(\frac{131}{132}\right)\)\(e\left(\frac{37}{66}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{73}{132}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{4}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(2557,a) \;\) at \(\;a = \) e.g. 2