Properties

Label 4029.343
Modulus $4029$
Conductor $1343$
Order $208$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, base_ring=CyclotomicField(208))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,13,8]))
 
pari: [g,chi] = znchar(Mod(343,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(1343\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(208\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1343}(343,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4029.ct

\(\chi_{4029}(58,\cdot)\) \(\chi_{4029}(61,\cdot)\) \(\chi_{4029}(91,\cdot)\) \(\chi_{4029}(112,\cdot)\) \(\chi_{4029}(148,\cdot)\) \(\chi_{4029}(175,\cdot)\) \(\chi_{4029}(199,\cdot)\) \(\chi_{4029}(295,\cdot)\) \(\chi_{4029}(328,\cdot)\) \(\chi_{4029}(343,\cdot)\) \(\chi_{4029}(385,\cdot)\) \(\chi_{4029}(436,\cdot)\) \(\chi_{4029}(466,\cdot)\) \(\chi_{4029}(532,\cdot)\) \(\chi_{4029}(568,\cdot)\) \(\chi_{4029}(622,\cdot)\) \(\chi_{4029}(649,\cdot)\) \(\chi_{4029}(673,\cdot)\) \(\chi_{4029}(703,\cdot)\) \(\chi_{4029}(772,\cdot)\) \(\chi_{4029}(802,\cdot)\) \(\chi_{4029}(805,\cdot)\) \(\chi_{4029}(823,\cdot)\) \(\chi_{4029}(847,\cdot)\) \(\chi_{4029}(940,\cdot)\) \(\chi_{4029}(1006,\cdot)\) \(\chi_{4029}(1009,\cdot)\) \(\chi_{4029}(1042,\cdot)\) \(\chi_{4029}(1060,\cdot)\) \(\chi_{4029}(1246,\cdot)\) ...

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: $\Q(\zeta_{208})$
Fixed field: Number field defined by a degree 208 polynomial (not computed)

Values on generators

\((2687,3556,3163)\) → \((1,e\left(\frac{1}{16}\right),e\left(\frac{1}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(343, a) \) \(1\)\(1\)\(e\left(\frac{3}{104}\right)\)\(e\left(\frac{3}{52}\right)\)\(e\left(\frac{145}{208}\right)\)\(e\left(\frac{151}{208}\right)\)\(e\left(\frac{9}{104}\right)\)\(e\left(\frac{151}{208}\right)\)\(e\left(\frac{11}{208}\right)\)\(e\left(\frac{29}{52}\right)\)\(e\left(\frac{157}{208}\right)\)\(e\left(\frac{3}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(343,a) \;\) at \(\;a = \) e.g. 2