Properties

Label 4000.2513
Modulus $4000$
Conductor $1000$
Order $100$
Real no
Primitive no
Minimal no
Parity odd

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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([0,50,39]))
 
pari: [g,chi] = znchar(Mod(2513,4000))
 

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 \(\chi_{1000}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4000.ct

\(\chi_{4000}(17,\cdot)\) \(\chi_{4000}(113,\cdot)\) \(\chi_{4000}(177,\cdot)\) \(\chi_{4000}(273,\cdot)\) \(\chi_{4000}(337,\cdot)\) \(\chi_{4000}(433,\cdot)\) \(\chi_{4000}(497,\cdot)\) \(\chi_{4000}(753,\cdot)\) \(\chi_{4000}(817,\cdot)\) \(\chi_{4000}(913,\cdot)\) \(\chi_{4000}(977,\cdot)\) \(\chi_{4000}(1073,\cdot)\) \(\chi_{4000}(1137,\cdot)\) \(\chi_{4000}(1233,\cdot)\) \(\chi_{4000}(1297,\cdot)\) \(\chi_{4000}(1553,\cdot)\) \(\chi_{4000}(1617,\cdot)\) \(\chi_{4000}(1713,\cdot)\) \(\chi_{4000}(1777,\cdot)\) \(\chi_{4000}(1873,\cdot)\) \(\chi_{4000}(1937,\cdot)\) \(\chi_{4000}(2033,\cdot)\) \(\chi_{4000}(2097,\cdot)\) \(\chi_{4000}(2353,\cdot)\) \(\chi_{4000}(2417,\cdot)\) \(\chi_{4000}(2513,\cdot)\) \(\chi_{4000}(2577,\cdot)\) \(\chi_{4000}(2673,\cdot)\) \(\chi_{4000}(2737,\cdot)\) \(\chi_{4000}(2833,\cdot)\) ...

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

\((2751,2501,1377)\) → \((1,-1,e\left(\frac{39}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(2513, a) \) \(-1\)\(1\)\(e\left(\frac{23}{100}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{23}{50}\right)\)\(e\left(\frac{7}{50}\right)\)\(e\left(\frac{71}{100}\right)\)\(e\left(\frac{47}{100}\right)\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{19}{50}\right)\)\(e\left(\frac{9}{100}\right)\)\(e\left(\frac{69}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(2513,a) \;\) at \(\;a = \) e.g. 2