Properties

Label 837.574
Modulus $837$
Conductor $837$
Order $45$
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(837, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([80,18]))
 
pari: [g,chi] = znchar(Mod(574,837))
 

Basic properties

Modulus: \(837\)
Conductor: \(837\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
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.cc

\(\chi_{837}(4,\cdot)\) \(\chi_{837}(16,\cdot)\) \(\chi_{837}(70,\cdot)\) \(\chi_{837}(97,\cdot)\) \(\chi_{837}(157,\cdot)\) \(\chi_{837}(202,\cdot)\) \(\chi_{837}(250,\cdot)\) \(\chi_{837}(256,\cdot)\) \(\chi_{837}(283,\cdot)\) \(\chi_{837}(295,\cdot)\) \(\chi_{837}(349,\cdot)\) \(\chi_{837}(376,\cdot)\) \(\chi_{837}(436,\cdot)\) \(\chi_{837}(481,\cdot)\) \(\chi_{837}(529,\cdot)\) \(\chi_{837}(535,\cdot)\) \(\chi_{837}(562,\cdot)\) \(\chi_{837}(574,\cdot)\) \(\chi_{837}(628,\cdot)\) \(\chi_{837}(655,\cdot)\) \(\chi_{837}(715,\cdot)\) \(\chi_{837}(760,\cdot)\) \(\chi_{837}(808,\cdot)\) \(\chi_{837}(814,\cdot)\)

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: 45.45.14871644565877379464047221413905669257566616785997555307108986009149096115256167660961155530887613343502969.1

Values on generators

\((218,406)\) → \((e\left(\frac{8}{9}\right),e\left(\frac{1}{5}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{31}{45}\right)\)\(e\left(\frac{17}{45}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{7}{45}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{23}{45}\right)\)\(e\left(\frac{34}{45}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 837 }(574,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{837}(574,\cdot)) = \sum_{r\in \Z/837\Z} \chi_{837}(574,r) e\left(\frac{2r}{837}\right) = -28.4479468519+5.2644391831i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 837 }(574,·),\chi_{ 837 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{837}(574,\cdot),\chi_{837}(1,\cdot)) = \sum_{r\in \Z/837\Z} \chi_{837}(574,r) \chi_{837}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 837 }(574,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{837}(574,·)) = \sum_{r \in \Z/837\Z} \chi_{837}(574,r) e\left(\frac{1 r + 2 r^{-1}}{837}\right) = -0.0 \)