Properties

Label 7803.5
Modulus $7803$
Conductor $7803$
Order $2448$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7803, base_ring=CyclotomicField(2448))
 
M = H._module
 
chi = DirichletCharacter(H, M([680,2061]))
 
pari: [g,chi] = znchar(Mod(5,7803))
 

Basic properties

Modulus: \(7803\)
Conductor: \(7803\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2448\)
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 7803.cg

\(\chi_{7803}(5,\cdot)\) \(\chi_{7803}(11,\cdot)\) \(\chi_{7803}(14,\cdot)\) \(\chi_{7803}(20,\cdot)\) \(\chi_{7803}(23,\cdot)\) \(\chi_{7803}(29,\cdot)\) \(\chi_{7803}(41,\cdot)\) \(\chi_{7803}(56,\cdot)\) \(\chi_{7803}(74,\cdot)\) \(\chi_{7803}(92,\cdot)\) \(\chi_{7803}(95,\cdot)\) \(\chi_{7803}(113,\cdot)\) \(\chi_{7803}(122,\cdot)\) \(\chi_{7803}(146,\cdot)\) \(\chi_{7803}(164,\cdot)\) \(\chi_{7803}(167,\cdot)\) \(\chi_{7803}(173,\cdot)\) \(\chi_{7803}(176,\cdot)\) \(\chi_{7803}(182,\cdot)\) \(\chi_{7803}(194,\cdot)\) \(\chi_{7803}(209,\cdot)\) \(\chi_{7803}(218,\cdot)\) \(\chi_{7803}(227,\cdot)\) \(\chi_{7803}(245,\cdot)\) \(\chi_{7803}(248,\cdot)\) \(\chi_{7803}(266,\cdot)\) \(\chi_{7803}(275,\cdot)\) \(\chi_{7803}(284,\cdot)\) \(\chi_{7803}(299,\cdot)\) \(\chi_{7803}(311,\cdot)\) ...

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: $\Q(\zeta_{2448})$
Fixed field: Number field defined by a degree 2448 polynomial (not computed)

Values on generators

\((2891,2026)\) → \((e\left(\frac{5}{18}\right),e\left(\frac{229}{272}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 7803 }(5, a) \) \(1\)\(1\)\(e\left(\frac{295}{1224}\right)\)\(e\left(\frac{295}{612}\right)\)\(e\left(\frac{457}{2448}\right)\)\(e\left(\frac{2303}{2448}\right)\)\(e\left(\frac{295}{408}\right)\)\(e\left(\frac{349}{816}\right)\)\(e\left(\frac{2387}{2448}\right)\)\(e\left(\frac{145}{612}\right)\)\(e\left(\frac{445}{2448}\right)\)\(e\left(\frac{295}{306}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7803 }(5,a) \;\) at \(\;a = \) e.g. 2