Properties

Label 441.22
Modulus $441$
Conductor $441$
Order $21$
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(441)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([7,12]))
 
pari: [g,chi] = znchar(Mod(22,441))
 

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 ]
 

Values on generators

\((344,199)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{4}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(1\)
value at e.g. 2

Related number fields

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 441 }(22,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{441}(22,\cdot)) = \sum_{r\in \Z/441\Z} \chi_{441}(22,r) e\left(\frac{2r}{441}\right) = -13.6696658521+15.9417764221i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 441 }(22,·),\chi_{ 441 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{441}(22,\cdot),\chi_{441}(1,\cdot)) = \sum_{r\in \Z/441\Z} \chi_{441}(22,r) \chi_{441}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 441 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{441}(22,·)) = \sum_{r \in \Z/441\Z} \chi_{441}(22,r) e\left(\frac{1 r + 2 r^{-1}}{441}\right) = 1.112254759+0.7583224546i \)