Properties

Label 575.7
Modulus $575$
Conductor $115$
Order $44$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,38]))
 
pari: [g,chi] = znchar(Mod(7,575))
 

Basic properties

Modulus: \(575\)
Conductor: \(115\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{115}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 575.r

\(\chi_{575}(7,\cdot)\) \(\chi_{575}(43,\cdot)\) \(\chi_{575}(57,\cdot)\) \(\chi_{575}(107,\cdot)\) \(\chi_{575}(132,\cdot)\) \(\chi_{575}(143,\cdot)\) \(\chi_{575}(157,\cdot)\) \(\chi_{575}(168,\cdot)\) \(\chi_{575}(182,\cdot)\) \(\chi_{575}(218,\cdot)\) \(\chi_{575}(268,\cdot)\) \(\chi_{575}(293,\cdot)\) \(\chi_{575}(318,\cdot)\) \(\chi_{575}(332,\cdot)\) \(\chi_{575}(343,\cdot)\) \(\chi_{575}(382,\cdot)\) \(\chi_{575}(457,\cdot)\) \(\chi_{575}(493,\cdot)\) \(\chi_{575}(543,\cdot)\) \(\chi_{575}(557,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: \(\Q(\zeta_{115})^+\)

Values on generators

\((277,51)\) → \((i,e\left(\frac{19}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 575 }(7, a) \) \(1\)\(1\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{37}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 575 }(7,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 575 }(7,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 575 }(7,·),\chi_{ 575 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 575 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2