Properties

Label 107.3
Modulus $107$
Conductor $107$
Order $53$
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(107, base_ring=CyclotomicField(106))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([70]))
 
pari: [g,chi] = znchar(Mod(3,107))
 

Basic properties

Modulus: \(107\)
Conductor: \(107\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(53\)
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 107.c

\(\chi_{107}(3,\cdot)\) \(\chi_{107}(4,\cdot)\) \(\chi_{107}(9,\cdot)\) \(\chi_{107}(10,\cdot)\) \(\chi_{107}(11,\cdot)\) \(\chi_{107}(12,\cdot)\) \(\chi_{107}(13,\cdot)\) \(\chi_{107}(14,\cdot)\) \(\chi_{107}(16,\cdot)\) \(\chi_{107}(19,\cdot)\) \(\chi_{107}(23,\cdot)\) \(\chi_{107}(25,\cdot)\) \(\chi_{107}(27,\cdot)\) \(\chi_{107}(29,\cdot)\) \(\chi_{107}(30,\cdot)\) \(\chi_{107}(33,\cdot)\) \(\chi_{107}(34,\cdot)\) \(\chi_{107}(35,\cdot)\) \(\chi_{107}(36,\cdot)\) \(\chi_{107}(37,\cdot)\) \(\chi_{107}(39,\cdot)\) \(\chi_{107}(40,\cdot)\) \(\chi_{107}(41,\cdot)\) \(\chi_{107}(42,\cdot)\) \(\chi_{107}(44,\cdot)\) \(\chi_{107}(47,\cdot)\) \(\chi_{107}(48,\cdot)\) \(\chi_{107}(49,\cdot)\) \(\chi_{107}(52,\cdot)\) \(\chi_{107}(53,\cdot)\) ...

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: $\Q(\zeta_{53})$
Fixed field: Number field defined by a degree 53 polynomial

Values on generators

\(2\) → \(e\left(\frac{35}{53}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{35}{53}\right)\)\(e\left(\frac{12}{53}\right)\)\(e\left(\frac{17}{53}\right)\)\(e\left(\frac{2}{53}\right)\)\(e\left(\frac{47}{53}\right)\)\(e\left(\frac{21}{53}\right)\)\(e\left(\frac{52}{53}\right)\)\(e\left(\frac{24}{53}\right)\)\(e\left(\frac{37}{53}\right)\)\(e\left(\frac{28}{53}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 107 }(3,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{107}(3,\cdot)) = \sum_{r\in \Z/107\Z} \chi_{107}(3,r) e\left(\frac{2r}{107}\right) = -10.2729683643+-1.2108348301i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 107 }(3,·),\chi_{ 107 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{107}(3,\cdot),\chi_{107}(1,\cdot)) = \sum_{r\in \Z/107\Z} \chi_{107}(3,r) \chi_{107}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 107 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{107}(3,·)) = \sum_{r \in \Z/107\Z} \chi_{107}(3,r) e\left(\frac{1 r + 2 r^{-1}}{107}\right) = -1.9062049857+3.457657981i \)