Properties

Label 4029.2416
Modulus $4029$
Conductor $1343$
Order $104$
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(104))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,91,40]))
 
pari: [g,chi] = znchar(Mod(2416,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(1343\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(104\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1343}(1073,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4029.cj

\(\chi_{4029}(100,\cdot)\) \(\chi_{4029}(196,\cdot)\) \(\chi_{4029}(223,\cdot)\) \(\chi_{4029}(247,\cdot)\) \(\chi_{4029}(304,\cdot)\) \(\chi_{4029}(433,\cdot)\) \(\chi_{4029}(457,\cdot)\) \(\chi_{4029}(484,\cdot)\) \(\chi_{4029}(733,\cdot)\) \(\chi_{4029}(757,\cdot)\) \(\chi_{4029}(763,\cdot)\) \(\chi_{4029}(808,\cdot)\) \(\chi_{4029}(994,\cdot)\) \(\chi_{4029}(1012,\cdot)\) \(\chi_{4029}(1045,\cdot)\) \(\chi_{4029}(1114,\cdot)\) \(\chi_{4029}(1171,\cdot)\) \(\chi_{4029}(1249,\cdot)\) \(\chi_{4029}(1351,\cdot)\) \(\chi_{4029}(1522,\cdot)\) \(\chi_{4029}(1681,\cdot)\) \(\chi_{4029}(1726,\cdot)\) \(\chi_{4029}(1759,\cdot)\) \(\chi_{4029}(1855,\cdot)\) \(\chi_{4029}(1879,\cdot)\) \(\chi_{4029}(1906,\cdot)\) \(\chi_{4029}(1963,\cdot)\) \(\chi_{4029}(2116,\cdot)\) \(\chi_{4029}(2185,\cdot)\) \(\chi_{4029}(2416,\cdot)\) ...

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: $\Q(\zeta_{104})$
Fixed field: Number field defined by a degree 104 polynomial (not computed)

Values on generators

\((2687,3556,3163)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{5}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(2416, a) \) \(1\)\(1\)\(e\left(\frac{41}{52}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{23}{104}\right)\)\(e\left(\frac{1}{104}\right)\)\(e\left(\frac{19}{52}\right)\)\(e\left(\frac{1}{104}\right)\)\(e\left(\frac{29}{104}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{83}{104}\right)\)\(e\left(\frac{2}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(2416,a) \;\) at \(\;a = \) e.g. 2