Properties

Label 605.16
Modulus $605$
Conductor $121$
Order $55$
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(605, base_ring=CyclotomicField(110))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,4]))
 
pari: [g,chi] = znchar(Mod(16,605))
 

Basic properties

Modulus: \(605\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(55\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{121}(16,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 605.s

\(\chi_{605}(16,\cdot)\) \(\chi_{605}(26,\cdot)\) \(\chi_{605}(31,\cdot)\) \(\chi_{605}(36,\cdot)\) \(\chi_{605}(71,\cdot)\) \(\chi_{605}(86,\cdot)\) \(\chi_{605}(91,\cdot)\) \(\chi_{605}(126,\cdot)\) \(\chi_{605}(136,\cdot)\) \(\chi_{605}(141,\cdot)\) \(\chi_{605}(146,\cdot)\) \(\chi_{605}(181,\cdot)\) \(\chi_{605}(191,\cdot)\) \(\chi_{605}(196,\cdot)\) \(\chi_{605}(201,\cdot)\) \(\chi_{605}(236,\cdot)\) \(\chi_{605}(246,\cdot)\) \(\chi_{605}(256,\cdot)\) \(\chi_{605}(291,\cdot)\) \(\chi_{605}(301,\cdot)\) \(\chi_{605}(306,\cdot)\) \(\chi_{605}(311,\cdot)\) \(\chi_{605}(346,\cdot)\) \(\chi_{605}(356,\cdot)\) \(\chi_{605}(361,\cdot)\) \(\chi_{605}(401,\cdot)\) \(\chi_{605}(411,\cdot)\) \(\chi_{605}(416,\cdot)\) \(\chi_{605}(421,\cdot)\) \(\chi_{605}(456,\cdot)\) ...

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

\((122,486)\) → \((1,e\left(\frac{2}{55}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(1\)\(1\)\(e\left(\frac{2}{55}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{4}{55}\right)\)\(e\left(\frac{13}{55}\right)\)\(e\left(\frac{14}{55}\right)\)\(e\left(\frac{6}{55}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{37}{55}\right)\)\(e\left(\frac{16}{55}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 55 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 605 }(16,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{605}(16,\cdot)) = \sum_{r\in \Z/605\Z} \chi_{605}(16,r) e\left(\frac{2r}{605}\right) = -5.2842676349+9.6476170925i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 605 }(16,·),\chi_{ 605 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{605}(16,\cdot),\chi_{605}(1,\cdot)) = \sum_{r\in \Z/605\Z} \chi_{605}(16,r) \chi_{605}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 605 }(16,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{605}(16,·)) = \sum_{r \in \Z/605\Z} \chi_{605}(16,r) e\left(\frac{1 r + 2 r^{-1}}{605}\right) = -0.0 \)