Properties

Label 175.103
Modulus $175$
Conductor $175$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(175)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,50]))
 
pari: [g,chi] = znchar(Mod(103,175))
 

Basic properties

Modulus: \(175\)
Conductor: \(175\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 175.x

\(\chi_{175}(3,\cdot)\) \(\chi_{175}(12,\cdot)\) \(\chi_{175}(17,\cdot)\) \(\chi_{175}(33,\cdot)\) \(\chi_{175}(38,\cdot)\) \(\chi_{175}(47,\cdot)\) \(\chi_{175}(52,\cdot)\) \(\chi_{175}(73,\cdot)\) \(\chi_{175}(87,\cdot)\) \(\chi_{175}(103,\cdot)\) \(\chi_{175}(108,\cdot)\) \(\chi_{175}(117,\cdot)\) \(\chi_{175}(122,\cdot)\) \(\chi_{175}(138,\cdot)\) \(\chi_{175}(152,\cdot)\) \(\chi_{175}(173,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((127,101)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(1\)\(1\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{1}{15}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 175 }(103,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{175}(103,\cdot)) = \sum_{r\in \Z/175\Z} \chi_{175}(103,r) e\left(\frac{2r}{175}\right) = 6.3855957765+11.5855153782i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 175 }(103,·),\chi_{ 175 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{175}(103,\cdot),\chi_{175}(1,\cdot)) = \sum_{r\in \Z/175\Z} \chi_{175}(103,r) \chi_{175}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 175 }(103,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{175}(103,·)) = \sum_{r \in \Z/175\Z} \chi_{175}(103,r) e\left(\frac{1 r + 2 r^{-1}}{175}\right) = -0.0 \)