Properties

Label 750.19
Modulus $750$
Conductor $125$
Order $50$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(750, base_ring=CyclotomicField(50))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9]))
 
pari: [g,chi] = znchar(Mod(19,750))
 

Basic properties

Modulus: \(750\)
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 \(\chi_{125}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 750.o

\(\chi_{750}(19,\cdot)\) \(\chi_{750}(79,\cdot)\) \(\chi_{750}(109,\cdot)\) \(\chi_{750}(139,\cdot)\) \(\chi_{750}(169,\cdot)\) \(\chi_{750}(229,\cdot)\) \(\chi_{750}(259,\cdot)\) \(\chi_{750}(289,\cdot)\) \(\chi_{750}(319,\cdot)\) \(\chi_{750}(379,\cdot)\) \(\chi_{750}(409,\cdot)\) \(\chi_{750}(439,\cdot)\) \(\chi_{750}(469,\cdot)\) \(\chi_{750}(529,\cdot)\) \(\chi_{750}(559,\cdot)\) \(\chi_{750}(589,\cdot)\) \(\chi_{750}(619,\cdot)\) \(\chi_{750}(679,\cdot)\) \(\chi_{750}(709,\cdot)\) \(\chi_{750}(739,\cdot)\)

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

\((251,127)\) → \((1,e\left(\frac{9}{50}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 750 }(19, a) \) \(1\)\(1\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{17}{25}\right)\)\(e\left(\frac{1}{50}\right)\)\(e\left(\frac{7}{50}\right)\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{29}{50}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{16}{25}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{23}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 750 }(19,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 750 }(19,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 750 }(19,·),\chi_{ 750 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 750 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2