Properties

Label 441.85
Modulus $441$
Conductor $441$
Order $21$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(441, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([14,12]))
 
pari: [g,chi] = znchar(Mod(85,441))
 

Basic properties

Modulus: \(441\)
Conductor: \(441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
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 441.ba

\(\chi_{441}(22,\cdot)\) \(\chi_{441}(43,\cdot)\) \(\chi_{441}(85,\cdot)\) \(\chi_{441}(106,\cdot)\) \(\chi_{441}(169,\cdot)\) \(\chi_{441}(211,\cdot)\) \(\chi_{441}(232,\cdot)\) \(\chi_{441}(274,\cdot)\) \(\chi_{441}(337,\cdot)\) \(\chi_{441}(358,\cdot)\) \(\chi_{441}(400,\cdot)\) \(\chi_{441}(421,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 21.21.60663096207149471029120391038078960708572561.1

Values on generators

\((344,199)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{2}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{1}{7}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 441 }(85,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 441 }(85,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 441 }(85,·),\chi_{ 441 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 441 }(85,·)) \;\) at \(\; a,b = \) e.g. 1,2