Properties

Label 175.18
Modulus $175$
Conductor $35$
Order $12$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(175, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,8]))
 
pari: [g,chi] = znchar(Mod(18,175))
 

Basic properties

Modulus: \(175\)
Conductor: \(35\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{35}(18,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 175.p

\(\chi_{175}(18,\cdot)\) \(\chi_{175}(32,\cdot)\) \(\chi_{175}(93,\cdot)\) \(\chi_{175}(107,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.0.11259376953125.1

Values on generators

\((127,101)\) → \((-i,e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(-1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(1\)\(i\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(i\)\(e\left(\frac{1}{3}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 175 }(18,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{175}(18,\cdot)) = \sum_{r\in \Z/175\Z} \chi_{175}(18,r) e\left(\frac{2r}{175}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 175 }(18,·),\chi_{ 175 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{175}(18,\cdot),\chi_{175}(1,\cdot)) = \sum_{r\in \Z/175\Z} \chi_{175}(18,r) \chi_{175}(1,1-r) = 5 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 175 }(18,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{175}(18,·)) = \sum_{r \in \Z/175\Z} \chi_{175}(18,r) e\left(\frac{1 r + 2 r^{-1}}{175}\right) = -0.0 \)