Properties

Label 103.10
Modulus $103$
Conductor $103$
Order $34$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(103)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([15]))
 
pari: [g,chi] = znchar(Mod(10,103))
 

Basic properties

Modulus: \(103\)
Conductor: \(103\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(34\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 103.f

\(\chi_{103}(3,\cdot)\) \(\chi_{103}(10,\cdot)\) \(\chi_{103}(22,\cdot)\) \(\chi_{103}(24,\cdot)\) \(\chi_{103}(27,\cdot)\) \(\chi_{103}(31,\cdot)\) \(\chi_{103}(37,\cdot)\) \(\chi_{103}(39,\cdot)\) \(\chi_{103}(42,\cdot)\) \(\chi_{103}(69,\cdot)\) \(\chi_{103}(73,\cdot)\) \(\chi_{103}(80,\cdot)\) \(\chi_{103}(89,\cdot)\) \(\chi_{103}(90,\cdot)\) \(\chi_{103}(94,\cdot)\) \(\chi_{103}(95,\cdot)\)

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

\(5\) → \(e\left(\frac{15}{34}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{7}{34}\right)\)\(e\left(\frac{14}{17}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{21}{34}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{4}{17}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{29}{34}\right)\)\(e\left(\frac{31}{34}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{17})\)
Fixed field: 34.0.2652335238355663972863781109929452800183143879582476922663961790823.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 103 }(10,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{103}(10,\cdot)) = \sum_{r\in \Z/103\Z} \chi_{103}(10,r) e\left(\frac{2r}{103}\right) = 6.6023519881+-7.7077200407i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 103 }(10,·),\chi_{ 103 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{103}(10,\cdot),\chi_{103}(1,\cdot)) = \sum_{r\in \Z/103\Z} \chi_{103}(10,r) \chi_{103}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 103 }(10,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{103}(10,·)) = \sum_{r \in \Z/103\Z} \chi_{103}(10,r) e\left(\frac{1 r + 2 r^{-1}}{103}\right) = 7.6779404581+-2.1845624083i \)