Properties

Label 7803.460
Modulus $7803$
Conductor $289$
Order $17$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7803, base_ring=CyclotomicField(34))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,2]))
 
pari: [g,chi] = znchar(Mod(460,7803))
 

Basic properties

Modulus: \(7803\)
Conductor: \(289\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(17\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{289}(171,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7803.r

\(\chi_{7803}(460,\cdot)\) \(\chi_{7803}(919,\cdot)\) \(\chi_{7803}(1378,\cdot)\) \(\chi_{7803}(1837,\cdot)\) \(\chi_{7803}(2296,\cdot)\) \(\chi_{7803}(2755,\cdot)\) \(\chi_{7803}(3214,\cdot)\) \(\chi_{7803}(3673,\cdot)\) \(\chi_{7803}(4132,\cdot)\) \(\chi_{7803}(4591,\cdot)\) \(\chi_{7803}(5050,\cdot)\) \(\chi_{7803}(5509,\cdot)\) \(\chi_{7803}(5968,\cdot)\) \(\chi_{7803}(6427,\cdot)\) \(\chi_{7803}(6886,\cdot)\) \(\chi_{7803}(7345,\cdot)\)

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: \(\Q(\zeta_{17})\)
Fixed field: Number field defined by a degree 17 polynomial

Values on generators

\((2891,2026)\) → \((1,e\left(\frac{1}{17}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 7803 }(460, a) \) \(1\)\(1\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{6}{17}\right)\)\(e\left(\frac{8}{17}\right)\)\(e\left(\frac{2}{17}\right)\)\(e\left(\frac{9}{17}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{6}{17}\right)\)\(e\left(\frac{9}{17}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{12}{17}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7803 }(460,a) \;\) at \(\;a = \) e.g. 2