Properties

Label 211.143
Modulus $211$
Conductor $211$
Order $35$
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(211, base_ring=CyclotomicField(70))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([32]))
 
pari: [g,chi] = znchar(Mod(143,211))
 

Basic properties

Modulus: \(211\)
Conductor: \(211\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(35\)
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 211.l

\(\chi_{211}(5,\cdot)\) \(\chi_{211}(11,\cdot)\) \(\chi_{211}(13,\cdot)\) \(\chi_{211}(25,\cdot)\) \(\chi_{211}(64,\cdot)\) \(\chi_{211}(65,\cdot)\) \(\chi_{211}(76,\cdot)\) \(\chi_{211}(79,\cdot)\) \(\chi_{211}(82,\cdot)\) \(\chi_{211}(87,\cdot)\) \(\chi_{211}(96,\cdot)\) \(\chi_{211}(109,\cdot)\) \(\chi_{211}(113,\cdot)\) \(\chi_{211}(114,\cdot)\) \(\chi_{211}(121,\cdot)\) \(\chi_{211}(122,\cdot)\) \(\chi_{211}(125,\cdot)\) \(\chi_{211}(143,\cdot)\) \(\chi_{211}(151,\cdot)\) \(\chi_{211}(169,\cdot)\) \(\chi_{211}(183,\cdot)\) \(\chi_{211}(184,\cdot)\) \(\chi_{211}(193,\cdot)\) \(\chi_{211}(203,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: 35.35.10607266494966666158512469468409065269845427817856998775062284008873509080731241.1

Values on generators

\(2\) → \(e\left(\frac{16}{35}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{23}{35}\right)\)\(e\left(\frac{32}{35}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{11}{35}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{35}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 211 }(143,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{211}(143,\cdot)) = \sum_{r\in \Z/211\Z} \chi_{211}(143,r) e\left(\frac{2r}{211}\right) = -6.8042818103+12.8336179251i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 211 }(143,·),\chi_{ 211 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{211}(143,\cdot),\chi_{211}(1,\cdot)) = \sum_{r\in \Z/211\Z} \chi_{211}(143,r) \chi_{211}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 211 }(143,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{211}(143,·)) = \sum_{r \in \Z/211\Z} \chi_{211}(143,r) e\left(\frac{1 r + 2 r^{-1}}{211}\right) = 2.7875426647+20.5784663187i \)