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Properties

Label 1078.135
Modulus $1078$
Conductor $539$
Order $105$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1078, base_ring=CyclotomicField(210)) M = H._module chi = DirichletCharacter(H, M([160,168]))
 
Copy content gp:[g,chi] = znchar(Mod(135, 1078))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1078.135");
 

Basic properties

Modulus: \(1078\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(539\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(105\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: no, induced from \(\chi_{539}(135,\cdot)\)
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1078.bc

\(\chi_{1078}(9,\cdot)\) \(\chi_{1078}(25,\cdot)\) \(\chi_{1078}(37,\cdot)\) \(\chi_{1078}(53,\cdot)\) \(\chi_{1078}(81,\cdot)\) \(\chi_{1078}(93,\cdot)\) \(\chi_{1078}(135,\cdot)\) \(\chi_{1078}(137,\cdot)\) \(\chi_{1078}(163,\cdot)\) \(\chi_{1078}(179,\cdot)\) \(\chi_{1078}(191,\cdot)\) \(\chi_{1078}(207,\cdot)\) \(\chi_{1078}(235,\cdot)\) \(\chi_{1078}(247,\cdot)\) \(\chi_{1078}(289,\cdot)\) \(\chi_{1078}(291,\cdot)\) \(\chi_{1078}(317,\cdot)\) \(\chi_{1078}(333,\cdot)\) \(\chi_{1078}(345,\cdot)\) \(\chi_{1078}(389,\cdot)\) \(\chi_{1078}(401,\cdot)\) \(\chi_{1078}(443,\cdot)\) \(\chi_{1078}(445,\cdot)\) \(\chi_{1078}(487,\cdot)\) \(\chi_{1078}(499,\cdot)\) \(\chi_{1078}(515,\cdot)\) \(\chi_{1078}(543,\cdot)\) \(\chi_{1078}(555,\cdot)\) \(\chi_{1078}(597,\cdot)\) \(\chi_{1078}(599,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 105 polynomial (not computed)

Values on generators

\((199,981)\) → \((e\left(\frac{16}{21}\right),e\left(\frac{4}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 1078 }(135, a) \) \(1\)\(1\)\(e\left(\frac{17}{105}\right)\)\(e\left(\frac{31}{105}\right)\)\(e\left(\frac{34}{105}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{26}{105}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{62}{105}\right)\)\(e\left(\frac{17}{35}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 1078 }(135,a) \;\) at \(\;a = \) e.g. 2