Properties

Label 8450.37
Modulus $8450$
Conductor $4225$
Order $780$
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(8450, base_ring=CyclotomicField(780))
 
M = H._module
 
chi = DirichletCharacter(H, M([351,755]))
 
pari: [g,chi] = znchar(Mod(37,8450))
 

Basic properties

Modulus: \(8450\)
Conductor: \(4225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(780\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4225}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8450.da

\(\chi_{8450}(37,\cdot)\) \(\chi_{8450}(123,\cdot)\) \(\chi_{8450}(137,\cdot)\) \(\chi_{8450}(167,\cdot)\) \(\chi_{8450}(223,\cdot)\) \(\chi_{8450}(253,\cdot)\) \(\chi_{8450}(267,\cdot)\) \(\chi_{8450}(297,\cdot)\) \(\chi_{8450}(353,\cdot)\) \(\chi_{8450}(383,\cdot)\) \(\chi_{8450}(397,\cdot)\) \(\chi_{8450}(483,\cdot)\) \(\chi_{8450}(513,\cdot)\) \(\chi_{8450}(527,\cdot)\) \(\chi_{8450}(613,\cdot)\) \(\chi_{8450}(687,\cdot)\) \(\chi_{8450}(773,\cdot)\) \(\chi_{8450}(787,\cdot)\) \(\chi_{8450}(817,\cdot)\) \(\chi_{8450}(873,\cdot)\) \(\chi_{8450}(903,\cdot)\) \(\chi_{8450}(917,\cdot)\) \(\chi_{8450}(947,\cdot)\) \(\chi_{8450}(1003,\cdot)\) \(\chi_{8450}(1047,\cdot)\) \(\chi_{8450}(1077,\cdot)\) \(\chi_{8450}(1133,\cdot)\) \(\chi_{8450}(1163,\cdot)\) \(\chi_{8450}(1177,\cdot)\) \(\chi_{8450}(1337,\cdot)\) ...

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: $\Q(\zeta_{780})$
Fixed field: Number field defined by a degree 780 polynomial (not computed)

Values on generators

\((677,3551)\) → \((e\left(\frac{9}{20}\right),e\left(\frac{151}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 8450 }(37, a) \) \(1\)\(1\)\(e\left(\frac{137}{780}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{137}{390}\right)\)\(e\left(\frac{701}{780}\right)\)\(e\left(\frac{133}{780}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{259}{260}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{137}{260}\right)\)\(e\left(\frac{241}{390}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8450 }(37,a) \;\) at \(\;a = \) e.g. 2