Properties

Label 4000.2287
Modulus $4000$
Conductor $1000$
Order $100$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(100))
 
M = H._module
 
chi = DirichletCharacter(H, M([50,50,29]))
 
pari: [g,chi] = znchar(Mod(2287,4000))
 

Basic properties

Modulus: \(4000\)
Conductor: \(1000\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1000}(787,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4000.cr

\(\chi_{4000}(47,\cdot)\) \(\chi_{4000}(303,\cdot)\) \(\chi_{4000}(367,\cdot)\) \(\chi_{4000}(463,\cdot)\) \(\chi_{4000}(527,\cdot)\) \(\chi_{4000}(623,\cdot)\) \(\chi_{4000}(687,\cdot)\) \(\chi_{4000}(783,\cdot)\) \(\chi_{4000}(847,\cdot)\) \(\chi_{4000}(1103,\cdot)\) \(\chi_{4000}(1167,\cdot)\) \(\chi_{4000}(1263,\cdot)\) \(\chi_{4000}(1327,\cdot)\) \(\chi_{4000}(1423,\cdot)\) \(\chi_{4000}(1487,\cdot)\) \(\chi_{4000}(1583,\cdot)\) \(\chi_{4000}(1647,\cdot)\) \(\chi_{4000}(1903,\cdot)\) \(\chi_{4000}(1967,\cdot)\) \(\chi_{4000}(2063,\cdot)\) \(\chi_{4000}(2127,\cdot)\) \(\chi_{4000}(2223,\cdot)\) \(\chi_{4000}(2287,\cdot)\) \(\chi_{4000}(2383,\cdot)\) \(\chi_{4000}(2447,\cdot)\) \(\chi_{4000}(2703,\cdot)\) \(\chi_{4000}(2767,\cdot)\) \(\chi_{4000}(2863,\cdot)\) \(\chi_{4000}(2927,\cdot)\) \(\chi_{4000}(3023,\cdot)\) ...

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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

\((2751,2501,1377)\) → \((-1,-1,e\left(\frac{29}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(2287, a) \) \(1\)\(1\)\(e\left(\frac{3}{100}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{1}{25}\right)\)\(e\left(\frac{81}{100}\right)\)\(e\left(\frac{17}{100}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{9}{50}\right)\)\(e\left(\frac{49}{100}\right)\)\(e\left(\frac{9}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(2287,a) \;\) at \(\;a = \) e.g. 2