Properties

Label 4000.1473
Modulus $4000$
Conductor $125$
Order $100$
Real no
Primitive no
Minimal yes
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([0,0,71]))
 
pari: [g,chi] = znchar(Mod(1473,4000))
 

Basic properties

Modulus: \(4000\)
Conductor: \(125\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{125}(98,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4000.cq

\(\chi_{4000}(33,\cdot)\) \(\chi_{4000}(97,\cdot)\) \(\chi_{4000}(353,\cdot)\) \(\chi_{4000}(417,\cdot)\) \(\chi_{4000}(513,\cdot)\) \(\chi_{4000}(577,\cdot)\) \(\chi_{4000}(673,\cdot)\) \(\chi_{4000}(737,\cdot)\) \(\chi_{4000}(833,\cdot)\) \(\chi_{4000}(897,\cdot)\) \(\chi_{4000}(1153,\cdot)\) \(\chi_{4000}(1217,\cdot)\) \(\chi_{4000}(1313,\cdot)\) \(\chi_{4000}(1377,\cdot)\) \(\chi_{4000}(1473,\cdot)\) \(\chi_{4000}(1537,\cdot)\) \(\chi_{4000}(1633,\cdot)\) \(\chi_{4000}(1697,\cdot)\) \(\chi_{4000}(1953,\cdot)\) \(\chi_{4000}(2017,\cdot)\) \(\chi_{4000}(2113,\cdot)\) \(\chi_{4000}(2177,\cdot)\) \(\chi_{4000}(2273,\cdot)\) \(\chi_{4000}(2337,\cdot)\) \(\chi_{4000}(2433,\cdot)\) \(\chi_{4000}(2497,\cdot)\) \(\chi_{4000}(2753,\cdot)\) \(\chi_{4000}(2817,\cdot)\) \(\chi_{4000}(2913,\cdot)\) \(\chi_{4000}(2977,\cdot)\) ...

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

\((2751,2501,1377)\) → \((1,1,e\left(\frac{71}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(1473, a) \) \(-1\)\(1\)\(e\left(\frac{97}{100}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{47}{50}\right)\)\(e\left(\frac{24}{25}\right)\)\(e\left(\frac{69}{100}\right)\)\(e\left(\frac{83}{100}\right)\)\(e\left(\frac{39}{50}\right)\)\(e\left(\frac{8}{25}\right)\)\(e\left(\frac{1}{100}\right)\)\(e\left(\frac{91}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(1473,a) \;\) at \(\;a = \) e.g. 2