Properties

Label 7803.16
Modulus $7803$
Conductor $7803$
Order $306$
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(306))
 
M = H._module
 
chi = DirichletCharacter(H, M([68,243]))
 
pari: [g,chi] = znchar(Mod(16,7803))
 

Basic properties

Modulus: \(7803\)
Conductor: \(7803\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(306\)
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.bx

\(\chi_{7803}(16,\cdot)\) \(\chi_{7803}(67,\cdot)\) \(\chi_{7803}(169,\cdot)\) \(\chi_{7803}(220,\cdot)\) \(\chi_{7803}(322,\cdot)\) \(\chi_{7803}(373,\cdot)\) \(\chi_{7803}(475,\cdot)\) \(\chi_{7803}(526,\cdot)\) \(\chi_{7803}(628,\cdot)\) \(\chi_{7803}(679,\cdot)\) \(\chi_{7803}(781,\cdot)\) \(\chi_{7803}(832,\cdot)\) \(\chi_{7803}(934,\cdot)\) \(\chi_{7803}(985,\cdot)\) \(\chi_{7803}(1087,\cdot)\) \(\chi_{7803}(1138,\cdot)\) \(\chi_{7803}(1240,\cdot)\) \(\chi_{7803}(1291,\cdot)\) \(\chi_{7803}(1393,\cdot)\) \(\chi_{7803}(1546,\cdot)\) \(\chi_{7803}(1597,\cdot)\) \(\chi_{7803}(1699,\cdot)\) \(\chi_{7803}(1750,\cdot)\) \(\chi_{7803}(1852,\cdot)\) \(\chi_{7803}(1903,\cdot)\) \(\chi_{7803}(2005,\cdot)\) \(\chi_{7803}(2056,\cdot)\) \(\chi_{7803}(2158,\cdot)\) \(\chi_{7803}(2209,\cdot)\) \(\chi_{7803}(2362,\cdot)\) ...

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: $\Q(\zeta_{153})$
Fixed field: Number field defined by a degree 306 polynomial (not computed)

Values on generators

\((2891,2026)\) → \((e\left(\frac{2}{9}\right),e\left(\frac{27}{34}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 7803 }(16, a) \) \(1\)\(1\)\(e\left(\frac{16}{153}\right)\)\(e\left(\frac{32}{153}\right)\)\(e\left(\frac{295}{306}\right)\)\(e\left(\frac{197}{306}\right)\)\(e\left(\frac{16}{51}\right)\)\(e\left(\frac{7}{102}\right)\)\(e\left(\frac{47}{306}\right)\)\(e\left(\frac{65}{153}\right)\)\(e\left(\frac{229}{306}\right)\)\(e\left(\frac{64}{153}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7803 }(16,a) \;\) at \(\;a = \) e.g. 2