Properties

Label 837.29
Modulus $837$
Conductor $837$
Order $90$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(837, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([5,27]))
 
pari: [g,chi] = znchar(Mod(29,837))
 

Basic properties

Modulus: \(837\)
Conductor: \(837\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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 837.ch

\(\chi_{837}(23,\cdot)\) \(\chi_{837}(29,\cdot)\) \(\chi_{837}(77,\cdot)\) \(\chi_{837}(122,\cdot)\) \(\chi_{837}(182,\cdot)\) \(\chi_{837}(209,\cdot)\) \(\chi_{837}(263,\cdot)\) \(\chi_{837}(275,\cdot)\) \(\chi_{837}(302,\cdot)\) \(\chi_{837}(308,\cdot)\) \(\chi_{837}(356,\cdot)\) \(\chi_{837}(401,\cdot)\) \(\chi_{837}(461,\cdot)\) \(\chi_{837}(488,\cdot)\) \(\chi_{837}(542,\cdot)\) \(\chi_{837}(554,\cdot)\) \(\chi_{837}(581,\cdot)\) \(\chi_{837}(587,\cdot)\) \(\chi_{837}(635,\cdot)\) \(\chi_{837}(680,\cdot)\) \(\chi_{837}(740,\cdot)\) \(\chi_{837}(767,\cdot)\) \(\chi_{837}(821,\cdot)\) \(\chi_{837}(833,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((218,406)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{3}{10}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{23}{90}\right)\)\(e\left(\frac{23}{45}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{13}{45}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{67}{90}\right)\)\(e\left(\frac{49}{90}\right)\)\(e\left(\frac{1}{45}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 837 }(29,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{837}(29,\cdot)) = \sum_{r\in \Z/837\Z} \chi_{837}(29,r) e\left(\frac{2r}{837}\right) = 14.1819182486+-25.2165262237i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 837 }(29,·),\chi_{ 837 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{837}(29,\cdot),\chi_{837}(1,\cdot)) = \sum_{r\in \Z/837\Z} \chi_{837}(29,r) \chi_{837}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 837 }(29,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{837}(29,·)) = \sum_{r \in \Z/837\Z} \chi_{837}(29,r) e\left(\frac{1 r + 2 r^{-1}}{837}\right) = 10.0594937384+10.4169107075i \)