Properties

Label 4010.13
Modulus $4010$
Conductor $2005$
Order $400$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4010, base_ring=CyclotomicField(400))
 
M = H._module
 
chi = DirichletCharacter(H, M([300,319]))
 
pari: [g,chi] = znchar(Mod(13,4010))
 

Basic properties

Modulus: \(4010\)
Conductor: \(2005\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(400\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2005}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4010.bz

\(\chi_{4010}(13,\cdot)\) \(\chi_{4010}(23,\cdot)\) \(\chi_{4010}(37,\cdot)\) \(\chi_{4010}(67,\cdot)\) \(\chi_{4010}(87,\cdot)\) \(\chi_{4010}(117,\cdot)\) \(\chi_{4010}(137,\cdot)\) \(\chi_{4010}(163,\cdot)\) \(\chi_{4010}(187,\cdot)\) \(\chi_{4010}(193,\cdot)\) \(\chi_{4010}(207,\cdot)\) \(\chi_{4010}(217,\cdot)\) \(\chi_{4010}(297,\cdot)\) \(\chi_{4010}(347,\cdot)\) \(\chi_{4010}(367,\cdot)\) \(\chi_{4010}(447,\cdot)\) \(\chi_{4010}(453,\cdot)\) \(\chi_{4010}(467,\cdot)\) \(\chi_{4010}(493,\cdot)\) \(\chi_{4010}(533,\cdot)\) \(\chi_{4010}(537,\cdot)\) \(\chi_{4010}(583,\cdot)\) \(\chi_{4010}(617,\cdot)\) \(\chi_{4010}(663,\cdot)\) \(\chi_{4010}(673,\cdot)\) \(\chi_{4010}(723,\cdot)\) \(\chi_{4010}(727,\cdot)\) \(\chi_{4010}(743,\cdot)\) \(\chi_{4010}(783,\cdot)\) \(\chi_{4010}(787,\cdot)\) ...

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: $\Q(\zeta_{400})$
Fixed field: Number field defined by a degree 400 polynomial (not computed)

Values on generators

\((2407,3211)\) → \((-i,e\left(\frac{319}{400}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4010 }(13, a) \) \(1\)\(1\)\(e\left(\frac{19}{400}\right)\)\(e\left(\frac{39}{200}\right)\)\(e\left(\frac{19}{200}\right)\)\(e\left(\frac{143}{200}\right)\)\(e\left(\frac{261}{400}\right)\)\(e\left(\frac{77}{400}\right)\)\(e\left(\frac{197}{400}\right)\)\(e\left(\frac{97}{400}\right)\)\(e\left(\frac{129}{400}\right)\)\(e\left(\frac{57}{400}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4010 }(13,a) \;\) at \(\;a = \) e.g. 2