Properties

Label 441.37
Modulus $441$
Conductor $49$
Order $21$
Real no
Primitive no
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([0,16]))
 
pari: [g,chi] = znchar(Mod(37,441))
 

Basic properties

Modulus: \(441\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 441.bb

\(\chi_{441}(37,\cdot)\) \(\chi_{441}(46,\cdot)\) \(\chi_{441}(100,\cdot)\) \(\chi_{441}(109,\cdot)\) \(\chi_{441}(163,\cdot)\) \(\chi_{441}(172,\cdot)\) \(\chi_{441}(235,\cdot)\) \(\chi_{441}(289,\cdot)\) \(\chi_{441}(298,\cdot)\) \(\chi_{441}(352,\cdot)\) \(\chi_{441}(415,\cdot)\) \(\chi_{441}(424,\cdot)\)

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

\((344,199)\) → \((1,e\left(\frac{16}{21}\right))\)

Values

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

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: \(\Q(\zeta_{49})^+\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 441 }(37,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{441}(37,\cdot)) = \sum_{r\in \Z/441\Z} \chi_{441}(37,r) e\left(\frac{2r}{441}\right) = -0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 441 }(37,·),\chi_{ 441 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{441}(37,\cdot),\chi_{441}(1,\cdot)) = \sum_{r\in \Z/441\Z} \chi_{441}(37,r) \chi_{441}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 441 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{441}(37,·)) = \sum_{r \in \Z/441\Z} \chi_{441}(37,r) e\left(\frac{1 r + 2 r^{-1}}{441}\right) = -0.0 \)