Properties

Label 6025.443
Modulus $6025$
Conductor $1205$
Order $240$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6025, base_ring=CyclotomicField(240))
 
M = H._module
 
chi = DirichletCharacter(H, M([180,109]))
 
pari: [g,chi] = znchar(Mod(443,6025))
 

Basic properties

Modulus: \(6025\)
Conductor: \(1205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(240\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1205}(443,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6025.jo

\(\chi_{6025}(132,\cdot)\) \(\chi_{6025}(157,\cdot)\) \(\chi_{6025}(368,\cdot)\) \(\chi_{6025}(443,\cdot)\) \(\chi_{6025}(568,\cdot)\) \(\chi_{6025}(793,\cdot)\) \(\chi_{6025}(807,\cdot)\) \(\chi_{6025}(832,\cdot)\) \(\chi_{6025}(1032,\cdot)\) \(\chi_{6025}(1068,\cdot)\) \(\chi_{6025}(1093,\cdot)\) \(\chi_{6025}(1143,\cdot)\) \(\chi_{6025}(1168,\cdot)\) \(\chi_{6025}(1257,\cdot)\) \(\chi_{6025}(1432,\cdot)\) \(\chi_{6025}(1632,\cdot)\) \(\chi_{6025}(1782,\cdot)\) \(\chi_{6025}(1818,\cdot)\) \(\chi_{6025}(1857,\cdot)\) \(\chi_{6025}(1882,\cdot)\) \(\chi_{6025}(2118,\cdot)\) \(\chi_{6025}(2182,\cdot)\) \(\chi_{6025}(2268,\cdot)\) \(\chi_{6025}(2318,\cdot)\) \(\chi_{6025}(2332,\cdot)\) \(\chi_{6025}(2368,\cdot)\) \(\chi_{6025}(2582,\cdot)\) \(\chi_{6025}(2682,\cdot)\) \(\chi_{6025}(2693,\cdot)\) \(\chi_{6025}(2707,\cdot)\) ...

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: $\Q(\zeta_{240})$
Fixed field: Number field defined by a degree 240 polynomial (not computed)

Values on generators

\((2652,2176)\) → \((-i,e\left(\frac{109}{240}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 6025 }(443, a) \) \(1\)\(1\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{109}{120}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{49}{240}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{17}{48}\right)\)\(e\left(\frac{119}{120}\right)\)\(e\left(\frac{143}{240}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6025 }(443,a) \;\) at \(\;a = \) e.g. 2