Properties

Label 10000.49
Modulus $10000$
Conductor $125$
Order $50$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10000, base_ring=CyclotomicField(50))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,47]))
 
pari: [g,chi] = znchar(Mod(49,10000))
 

Basic properties

Modulus: \(10000\)
Conductor: \(125\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{125}(84,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 10000.bu

\(\chi_{10000}(49,\cdot)\) \(\chi_{10000}(449,\cdot)\) \(\chi_{10000}(849,\cdot)\) \(\chi_{10000}(1649,\cdot)\) \(\chi_{10000}(2049,\cdot)\) \(\chi_{10000}(2449,\cdot)\) \(\chi_{10000}(2849,\cdot)\) \(\chi_{10000}(3649,\cdot)\) \(\chi_{10000}(4049,\cdot)\) \(\chi_{10000}(4449,\cdot)\) \(\chi_{10000}(4849,\cdot)\) \(\chi_{10000}(5649,\cdot)\) \(\chi_{10000}(6049,\cdot)\) \(\chi_{10000}(6449,\cdot)\) \(\chi_{10000}(6849,\cdot)\) \(\chi_{10000}(7649,\cdot)\) \(\chi_{10000}(8049,\cdot)\) \(\chi_{10000}(8449,\cdot)\) \(\chi_{10000}(8849,\cdot)\) \(\chi_{10000}(9649,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Values on generators

\((8751,2501,9377)\) → \((1,1,e\left(\frac{47}{50}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 10000 }(49, a) \) \(1\)\(1\)\(e\left(\frac{29}{50}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{11}{25}\right)\)\(e\left(\frac{33}{50}\right)\)\(e\left(\frac{31}{50}\right)\)\(e\left(\frac{23}{25}\right)\)\(e\left(\frac{12}{25}\right)\)\(e\left(\frac{7}{50}\right)\)\(e\left(\frac{37}{50}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 10000 }(49,a) \;\) at \(\;a = \) e.g. 2