Properties

Label 4005.224
Modulus $4005$
Conductor $1335$
Order $88$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4005, base_ring=CyclotomicField(88))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,44,73]))
 
pari: [g,chi] = znchar(Mod(224,4005))
 

Basic properties

Modulus: \(4005\)
Conductor: \(1335\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(88\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1335}(224,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4005.dg

\(\chi_{4005}(224,\cdot)\) \(\chi_{4005}(359,\cdot)\) \(\chi_{4005}(404,\cdot)\) \(\chi_{4005}(629,\cdot)\) \(\chi_{4005}(674,\cdot)\) \(\chi_{4005}(719,\cdot)\) \(\chi_{4005}(944,\cdot)\) \(\chi_{4005}(1124,\cdot)\) \(\chi_{4005}(1259,\cdot)\) \(\chi_{4005}(1304,\cdot)\) \(\chi_{4005}(1349,\cdot)\) \(\chi_{4005}(1394,\cdot)\) \(\chi_{4005}(1439,\cdot)\) \(\chi_{4005}(1484,\cdot)\) \(\chi_{4005}(1574,\cdot)\) \(\chi_{4005}(1664,\cdot)\) \(\chi_{4005}(1754,\cdot)\) \(\chi_{4005}(1799,\cdot)\) \(\chi_{4005}(1934,\cdot)\) \(\chi_{4005}(2024,\cdot)\) \(\chi_{4005}(2159,\cdot)\) \(\chi_{4005}(2249,\cdot)\) \(\chi_{4005}(2384,\cdot)\) \(\chi_{4005}(2429,\cdot)\) \(\chi_{4005}(2519,\cdot)\) \(\chi_{4005}(2609,\cdot)\) \(\chi_{4005}(2699,\cdot)\) \(\chi_{4005}(2744,\cdot)\) \(\chi_{4005}(2789,\cdot)\) \(\chi_{4005}(2834,\cdot)\) ...

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: $\Q(\zeta_{88})$
Fixed field: Number field defined by a degree 88 polynomial

Values on generators

\((3116,802,181)\) → \((-1,-1,e\left(\frac{73}{88}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 4005 }(224, a) \) \(1\)\(1\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{61}{88}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{51}{88}\right)\)\(e\left(\frac{85}{88}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{3}{88}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4005 }(224,a) \;\) at \(\;a = \) e.g. 2