Properties

Label 4005.7
Modulus $4005$
Conductor $4005$
Order $264$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4005, base_ring=CyclotomicField(264))
 
M = H._module
 
chi = DirichletCharacter(H, M([176,66,243]))
 
pari: [g,chi] = znchar(Mod(7,4005))
 

Basic properties

Modulus: \(4005\)
Conductor: \(4005\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(264\)
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 4005.ed

\(\chi_{4005}(7,\cdot)\) \(\chi_{4005}(13,\cdot)\) \(\chi_{4005}(58,\cdot)\) \(\chi_{4005}(112,\cdot)\) \(\chi_{4005}(148,\cdot)\) \(\chi_{4005}(193,\cdot)\) \(\chi_{4005}(202,\cdot)\) \(\chi_{4005}(238,\cdot)\) \(\chi_{4005}(412,\cdot)\) \(\chi_{4005}(418,\cdot)\) \(\chi_{4005}(508,\cdot)\) \(\chi_{4005}(553,\cdot)\) \(\chi_{4005}(562,\cdot)\) \(\chi_{4005}(637,\cdot)\) \(\chi_{4005}(763,\cdot)\) \(\chi_{4005}(787,\cdot)\) \(\chi_{4005}(862,\cdot)\) \(\chi_{4005}(1012,\cdot)\) \(\chi_{4005}(1033,\cdot)\) \(\chi_{4005}(1138,\cdot)\) \(\chi_{4005}(1183,\cdot)\) \(\chi_{4005}(1222,\cdot)\) \(\chi_{4005}(1273,\cdot)\) \(\chi_{4005}(1312,\cdot)\) \(\chi_{4005}(1348,\cdot)\) \(\chi_{4005}(1393,\cdot)\) \(\chi_{4005}(1417,\cdot)\) \(\chi_{4005}(1447,\cdot)\) \(\chi_{4005}(1453,\cdot)\) \(\chi_{4005}(1483,\cdot)\) ...

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: $\Q(\zeta_{264})$
Fixed field: Number field defined by a degree 264 polynomial (not computed)

Values on generators

\((3116,802,181)\) → \((e\left(\frac{2}{3}\right),i,e\left(\frac{81}{88}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 4005 }(7, a) \) \(1\)\(1\)\(e\left(\frac{85}{132}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{125}{264}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{67}{264}\right)\)\(e\left(\frac{31}{264}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{63}{88}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4005 }(7,a) \;\) at \(\;a = \) e.g. 2