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Properties

Label 317417.248
Modulus $317417$
Conductor $317417$
Order $38584$
Real no
Primitive yes
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(317417, base_ring=CyclotomicField(38584)) M = H._module chi = DirichletCharacter(H, M([20160,29627]))
 
Copy content gp:[g,chi] = znchar(Mod(248, 317417))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("317417.248");
 

Basic properties

Modulus: \(317417\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(317417\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(38584\)
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: yes
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 317417.fj

\(\chi_{317417}(13,\cdot)\) \(\chi_{317417}(36,\cdot)\) \(\chi_{317417}(63,\cdot)\) \(\chi_{317417}(77,\cdot)\) \(\chi_{317417}(100,\cdot)\) \(\chi_{317417}(102,\cdot)\) \(\chi_{317417}(122,\cdot)\) \(\chi_{317417}(174,\cdot)\) \(\chi_{317417}(175,\cdot)\) \(\chi_{317417}(195,\cdot)\) \(\chi_{317417}(201,\cdot)\) \(\chi_{317417}(248,\cdot)\) \(\chi_{317417}(278,\cdot)\) \(\chi_{317417}(289,\cdot)\) \(\chi_{317417}(314,\cdot)\) \(\chi_{317417}(328,\cdot)\) \(\chi_{317417}(364,\cdot)\) \(\chi_{317417}(365,\cdot)\) \(\chi_{317417}(439,\cdot)\) \(\chi_{317417}(493,\cdot)\) \(\chi_{317417}(513,\cdot)\) \(\chi_{317417}(524,\cdot)\) \(\chi_{317417}(540,\cdot)\) \(\chi_{317417}(543,\cdot)\) \(\chi_{317417}(554,\cdot)\) \(\chi_{317417}(574,\cdot)\) \(\chi_{317417}(576,\cdot)\) \(\chi_{317417}(596,\cdot)\) \(\chi_{317417}(627,\cdot)\) \(\chi_{317417}(652,\cdot)\) ...

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: $\Q(\zeta_{38584})$
Fixed field: Number field defined by a degree 38584 polynomial (not computed)

Values on generators

\((297756,39327)\) → \((e\left(\frac{360}{689}\right),e\left(\frac{43}{56}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 317417 }(248, a) \) \(1\)\(1\)\(e\left(\frac{7107}{9646}\right)\)\(e\left(\frac{6891}{38584}\right)\)\(e\left(\frac{2284}{4823}\right)\)\(e\left(\frac{14809}{38584}\right)\)\(e\left(\frac{35319}{38584}\right)\)\(e\left(\frac{2754}{4823}\right)\)\(e\left(\frac{2029}{9646}\right)\)\(e\left(\frac{6891}{19292}\right)\)\(e\left(\frac{4653}{38584}\right)\)\(e\left(\frac{1343}{19292}\right)\)
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)
 
\( \chi_{ 317417 }(248,a) \;\) at \(\;a = \) e.g. 2