Properties

Label 207.13
Modulus $207$
Conductor $207$
Order $33$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(207, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([22,42]))
 
pari: [g,chi] = znchar(Mod(13,207))
 

Basic properties

Modulus: \(207\)
Conductor: \(207\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
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 207.m

\(\chi_{207}(4,\cdot)\) \(\chi_{207}(13,\cdot)\) \(\chi_{207}(16,\cdot)\) \(\chi_{207}(25,\cdot)\) \(\chi_{207}(31,\cdot)\) \(\chi_{207}(49,\cdot)\) \(\chi_{207}(52,\cdot)\) \(\chi_{207}(58,\cdot)\) \(\chi_{207}(85,\cdot)\) \(\chi_{207}(94,\cdot)\) \(\chi_{207}(121,\cdot)\) \(\chi_{207}(124,\cdot)\) \(\chi_{207}(133,\cdot)\) \(\chi_{207}(142,\cdot)\) \(\chi_{207}(151,\cdot)\) \(\chi_{207}(169,\cdot)\) \(\chi_{207}(187,\cdot)\) \(\chi_{207}(193,\cdot)\) \(\chi_{207}(196,\cdot)\) \(\chi_{207}(202,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: 33.33.70011645999218458416472683122408534303895571350166174758601569.1

Values on generators

\((47,28)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{11}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 207 }(13, a) \) \(1\)\(1\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{14}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 207 }(13,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 207 }(13,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 207 }(13,·),\chi_{ 207 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 207 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2