Properties

Label 4025.1461
Modulus $4025$
Conductor $4025$
Order $330$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4025, base_ring=CyclotomicField(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([264,275,300]))
 
pari: [g,chi] = znchar(Mod(1461,4025))
 

Basic properties

Modulus: \(4025\)
Conductor: \(4025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4025.di

\(\chi_{4025}(31,\cdot)\) \(\chi_{4025}(96,\cdot)\) \(\chi_{4025}(131,\cdot)\) \(\chi_{4025}(236,\cdot)\) \(\chi_{4025}(271,\cdot)\) \(\chi_{4025}(311,\cdot)\) \(\chi_{4025}(381,\cdot)\) \(\chi_{4025}(416,\cdot)\) \(\chi_{4025}(446,\cdot)\) \(\chi_{4025}(486,\cdot)\) \(\chi_{4025}(556,\cdot)\) \(\chi_{4025}(591,\cdot)\) \(\chi_{4025}(656,\cdot)\) \(\chi_{4025}(696,\cdot)\) \(\chi_{4025}(731,\cdot)\) \(\chi_{4025}(761,\cdot)\) \(\chi_{4025}(831,\cdot)\) \(\chi_{4025}(836,\cdot)\) \(\chi_{4025}(906,\cdot)\) \(\chi_{4025}(936,\cdot)\) \(\chi_{4025}(1041,\cdot)\) \(\chi_{4025}(1116,\cdot)\) \(\chi_{4025}(1181,\cdot)\) \(\chi_{4025}(1186,\cdot)\) \(\chi_{4025}(1221,\cdot)\) \(\chi_{4025}(1291,\cdot)\) \(\chi_{4025}(1361,\cdot)\) \(\chi_{4025}(1396,\cdot)\) \(\chi_{4025}(1461,\cdot)\) \(\chi_{4025}(1531,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((2577,1151,3501)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{5}{6}\right),e\left(\frac{10}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(1461, a) \) \(-1\)\(1\)\(e\left(\frac{47}{165}\right)\)\(e\left(\frac{323}{330}\right)\)\(e\left(\frac{94}{165}\right)\)\(e\left(\frac{29}{110}\right)\)\(e\left(\frac{47}{55}\right)\)\(e\left(\frac{158}{165}\right)\)\(e\left(\frac{52}{165}\right)\)\(e\left(\frac{181}{330}\right)\)\(e\left(\frac{47}{110}\right)\)\(e\left(\frac{23}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(1461,a) \;\) at \(\;a = \) e.g. 2