Properties

Label 101.37
Modulus $101$
Conductor $101$
Order $25$
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(101, base_ring=CyclotomicField(50))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([28]))
 
pari: [g,chi] = znchar(Mod(37,101))
 

Basic properties

Modulus: \(101\)
Conductor: \(101\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(25\)
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 101.g

\(\chi_{101}(5,\cdot)\) \(\chi_{101}(16,\cdot)\) \(\chi_{101}(19,\cdot)\) \(\chi_{101}(24,\cdot)\) \(\chi_{101}(25,\cdot)\) \(\chi_{101}(31,\cdot)\) \(\chi_{101}(37,\cdot)\) \(\chi_{101}(52,\cdot)\) \(\chi_{101}(54,\cdot)\) \(\chi_{101}(56,\cdot)\) \(\chi_{101}(58,\cdot)\) \(\chi_{101}(68,\cdot)\) \(\chi_{101}(71,\cdot)\) \(\chi_{101}(78,\cdot)\) \(\chi_{101}(79,\cdot)\) \(\chi_{101}(80,\cdot)\) \(\chi_{101}(81,\cdot)\) \(\chi_{101}(88,\cdot)\) \(\chi_{101}(92,\cdot)\) \(\chi_{101}(97,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: 25.25.1269734648531914468903714880493455422104626762401.1

Values on generators

\(2\) → \(e\left(\frac{14}{25}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{14}{25}\right)\)\(e\left(\frac{16}{25}\right)\)\(e\left(\frac{3}{25}\right)\)\(e\left(\frac{11}{25}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{25}\right)\)\(e\left(\frac{17}{25}\right)\)\(e\left(\frac{7}{25}\right)\)\(1\)\(e\left(\frac{7}{25}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 101 }(37,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{101}(37,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(37,r) e\left(\frac{2r}{101}\right) = -6.9589585909+7.2507168839i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 101 }(37,·),\chi_{ 101 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{101}(37,\cdot),\chi_{101}(1,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(37,r) \chi_{101}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 101 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{101}(37,·)) = \sum_{r \in \Z/101\Z} \chi_{101}(37,r) e\left(\frac{1 r + 2 r^{-1}}{101}\right) = -2.006020647+10.515928499i \)