Properties

Label 1045.148
Modulus $1045$
Conductor $1045$
Order $180$
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(1045, base_ring=CyclotomicField(180))
 
M = H._module
 
chi = DirichletCharacter(H, M([135,72,110]))
 
pari: [g,chi] = znchar(Mod(148,1045))
 

Basic properties

Modulus: \(1045\)
Conductor: \(1045\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
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 1045.cr

\(\chi_{1045}(3,\cdot)\) \(\chi_{1045}(48,\cdot)\) \(\chi_{1045}(53,\cdot)\) \(\chi_{1045}(97,\cdot)\) \(\chi_{1045}(108,\cdot)\) \(\chi_{1045}(147,\cdot)\) \(\chi_{1045}(148,\cdot)\) \(\chi_{1045}(192,\cdot)\) \(\chi_{1045}(203,\cdot)\) \(\chi_{1045}(212,\cdot)\) \(\chi_{1045}(223,\cdot)\) \(\chi_{1045}(257,\cdot)\) \(\chi_{1045}(262,\cdot)\) \(\chi_{1045}(268,\cdot)\) \(\chi_{1045}(317,\cdot)\) \(\chi_{1045}(333,\cdot)\) \(\chi_{1045}(357,\cdot)\) \(\chi_{1045}(383,\cdot)\) \(\chi_{1045}(412,\cdot)\) \(\chi_{1045}(432,\cdot)\) \(\chi_{1045}(433,\cdot)\) \(\chi_{1045}(477,\cdot)\) \(\chi_{1045}(478,\cdot)\) \(\chi_{1045}(488,\cdot)\) \(\chi_{1045}(542,\cdot)\) \(\chi_{1045}(553,\cdot)\) \(\chi_{1045}(592,\cdot)\) \(\chi_{1045}(603,\cdot)\) \(\chi_{1045}(642,\cdot)\) \(\chi_{1045}(687,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((837,761,496)\) → \((-i,e\left(\frac{2}{5}\right),e\left(\frac{11}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(148, a) \) \(1\)\(1\)\(e\left(\frac{137}{180}\right)\)\(e\left(\frac{71}{180}\right)\)\(e\left(\frac{47}{90}\right)\)\(e\left(\frac{7}{45}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{71}{90}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{127}{180}\right)\)\(e\left(\frac{44}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(148,a) \;\) at \(\;a = \) e.g. 2