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Properties

Label 38437.2762
Modulus $38437$
Conductor $38437$
Order $408$
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(38437, base_ring=CyclotomicField(408)) M = H._module chi = DirichletCharacter(H, M([272,159,136]))
 
Copy content gp:[g,chi] = znchar(Mod(2762, 38437))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("38437.2762");
 

Basic properties

Modulus: \(38437\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(38437\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(408\)
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 38437.jw

\(\chi_{38437}(501,\cdot)\) \(\chi_{38437}(695,\cdot)\) \(\chi_{38437}(767,\cdot)\) \(\chi_{38437}(961,\cdot)\) \(\chi_{38437}(1698,\cdot)\) \(\chi_{38437}(1759,\cdot)\) \(\chi_{38437}(1964,\cdot)\) \(\chi_{38437}(2025,\cdot)\) \(\chi_{38437}(2762,\cdot)\) \(\chi_{38437}(2956,\cdot)\) \(\chi_{38437}(3028,\cdot)\) \(\chi_{38437}(3222,\cdot)\) \(\chi_{38437}(3959,\cdot)\) \(\chi_{38437}(4020,\cdot)\) \(\chi_{38437}(4286,\cdot)\) \(\chi_{38437}(5217,\cdot)\) \(\chi_{38437}(5289,\cdot)\) \(\chi_{38437}(5483,\cdot)\) \(\chi_{38437}(6220,\cdot)\) \(\chi_{38437}(6281,\cdot)\) \(\chi_{38437}(6486,\cdot)\) \(\chi_{38437}(6547,\cdot)\) \(\chi_{38437}(7284,\cdot)\) \(\chi_{38437}(7478,\cdot)\) \(\chi_{38437}(7550,\cdot)\) \(\chi_{38437}(7744,\cdot)\) \(\chi_{38437}(8481,\cdot)\) \(\chi_{38437}(8542,\cdot)\) \(\chi_{38437}(8747,\cdot)\) \(\chi_{38437}(8808,\cdot)\) ...

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: $\Q(\zeta_{408})$
Fixed field: Number field defined by a degree 408 polynomial (not computed)

Values on generators

\((16474,30059,34392)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{53}{136}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 38437 }(2762, a) \) \(1\)\(1\)\(e\left(\frac{145}{204}\right)\)\(e\left(\frac{53}{136}\right)\)\(e\left(\frac{43}{102}\right)\)\(e\left(\frac{371}{408}\right)\)\(e\left(\frac{41}{408}\right)\)\(e\left(\frac{9}{68}\right)\)\(e\left(\frac{53}{68}\right)\)\(e\left(\frac{253}{408}\right)\)\(e\left(\frac{257}{408}\right)\)\(e\left(\frac{331}{408}\right)\)
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)
 
\( \chi_{ 38437 }(2762,a) \;\) at \(\;a = \) e.g. 2