Properties

Label 4025.513
Modulus $4025$
Conductor $4025$
Order $660$
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(4025, base_ring=CyclotomicField(660))
 
M = H._module
 
chi = DirichletCharacter(H, M([627,220,570]))
 
pari: [g,chi] = znchar(Mod(513,4025))
 

Basic properties

Modulus: \(4025\)
Conductor: \(4025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(660\)
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 4025.dr

\(\chi_{4025}(37,\cdot)\) \(\chi_{4025}(53,\cdot)\) \(\chi_{4025}(67,\cdot)\) \(\chi_{4025}(88,\cdot)\) \(\chi_{4025}(102,\cdot)\) \(\chi_{4025}(158,\cdot)\) \(\chi_{4025}(172,\cdot)\) \(\chi_{4025}(198,\cdot)\) \(\chi_{4025}(212,\cdot)\) \(\chi_{4025}(228,\cdot)\) \(\chi_{4025}(247,\cdot)\) \(\chi_{4025}(263,\cdot)\) \(\chi_{4025}(333,\cdot)\) \(\chi_{4025}(352,\cdot)\) \(\chi_{4025}(373,\cdot)\) \(\chi_{4025}(387,\cdot)\) \(\chi_{4025}(408,\cdot)\) \(\chi_{4025}(452,\cdot)\) \(\chi_{4025}(513,\cdot)\) \(\chi_{4025}(527,\cdot)\) \(\chi_{4025}(548,\cdot)\) \(\chi_{4025}(562,\cdot)\) \(\chi_{4025}(592,\cdot)\) \(\chi_{4025}(613,\cdot)\) \(\chi_{4025}(688,\cdot)\) \(\chi_{4025}(697,\cdot)\) \(\chi_{4025}(723,\cdot)\) \(\chi_{4025}(753,\cdot)\) \(\chi_{4025}(802,\cdot)\) \(\chi_{4025}(842,\cdot)\) ...

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: $\Q(\zeta_{660})$
Fixed field: Number field defined by a degree 660 polynomial (not computed)

Values on generators

\((2577,1151,3501)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{1}{3}\right),e\left(\frac{19}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(513, a) \) \(1\)\(1\)\(e\left(\frac{227}{660}\right)\)\(e\left(\frac{529}{660}\right)\)\(e\left(\frac{227}{330}\right)\)\(e\left(\frac{8}{55}\right)\)\(e\left(\frac{7}{220}\right)\)\(e\left(\frac{199}{330}\right)\)\(e\left(\frac{101}{330}\right)\)\(e\left(\frac{323}{660}\right)\)\(e\left(\frac{31}{220}\right)\)\(e\left(\frac{62}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(513,a) \;\) at \(\;a = \) e.g. 2