Properties

Label 4029.31
Modulus $4029$
Conductor $1343$
Order $624$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, base_ring=CyclotomicField(624))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,351,448]))
 
pari: [g,chi] = znchar(Mod(31,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(1343\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(624\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1343}(31,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4029.da

\(\chi_{4029}(31,\cdot)\) \(\chi_{4029}(40,\cdot)\) \(\chi_{4029}(73,\cdot)\) \(\chi_{4029}(88,\cdot)\) \(\chi_{4029}(124,\cdot)\) \(\chi_{4029}(130,\cdot)\) \(\chi_{4029}(160,\cdot)\) \(\chi_{4029}(163,\cdot)\) \(\chi_{4029}(184,\cdot)\) \(\chi_{4029}(190,\cdot)\) \(\chi_{4029}(241,\cdot)\) \(\chi_{4029}(250,\cdot)\) \(\chi_{4029}(262,\cdot)\) \(\chi_{4029}(277,\cdot)\) \(\chi_{4029}(286,\cdot)\) \(\chi_{4029}(313,\cdot)\) \(\chi_{4029}(352,\cdot)\) \(\chi_{4029}(367,\cdot)\) \(\chi_{4029}(388,\cdot)\) \(\chi_{4029}(397,\cdot)\) \(\chi_{4029}(415,\cdot)\) \(\chi_{4029}(439,\cdot)\) \(\chi_{4029}(445,\cdot)\) \(\chi_{4029}(487,\cdot)\) \(\chi_{4029}(490,\cdot)\) \(\chi_{4029}(499,\cdot)\) \(\chi_{4029}(505,\cdot)\) \(\chi_{4029}(547,\cdot)\) \(\chi_{4029}(550,\cdot)\) \(\chi_{4029}(589,\cdot)\) ...

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: $\Q(\zeta_{624})$
Fixed field: Number field defined by a degree 624 polynomial (not computed)

Values on generators

\((2687,3556,3163)\) → \((1,e\left(\frac{9}{16}\right),e\left(\frac{28}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(31, a) \) \(-1\)\(1\)\(e\left(\frac{233}{312}\right)\)\(e\left(\frac{77}{156}\right)\)\(e\left(\frac{203}{624}\right)\)\(e\left(\frac{149}{624}\right)\)\(e\left(\frac{25}{104}\right)\)\(e\left(\frac{15}{208}\right)\)\(e\left(\frac{473}{624}\right)\)\(e\left(\frac{103}{156}\right)\)\(e\left(\frac{205}{208}\right)\)\(e\left(\frac{77}{78}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(31,a) \;\) at \(\;a = \) e.g. 2