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Properties

Label 3107.425
Modulus $3107$
Conductor $3107$
Order $357$
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(3107, base_ring=CyclotomicField(714)) M = H._module chi = DirichletCharacter(H, M([476,270]))
 
Copy content gp:[g,chi] = znchar(Mod(425, 3107))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3107.425");
 

Basic properties

Modulus: \(3107\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3107\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(357\)
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 3107.bo

\(\chi_{3107}(3,\cdot)\) \(\chi_{3107}(9,\cdot)\) \(\chi_{3107}(16,\cdot)\) \(\chi_{3107}(29,\cdot)\) \(\chi_{3107}(48,\cdot)\) \(\chi_{3107}(55,\cdot)\) \(\chi_{3107}(61,\cdot)\) \(\chi_{3107}(68,\cdot)\) \(\chi_{3107}(81,\cdot)\) \(\chi_{3107}(87,\cdot)\) \(\chi_{3107}(113,\cdot)\) \(\chi_{3107}(120,\cdot)\) \(\chi_{3107}(133,\cdot)\) \(\chi_{3107}(165,\cdot)\) \(\chi_{3107}(198,\cdot)\) \(\chi_{3107}(204,\cdot)\) \(\chi_{3107}(243,\cdot)\) \(\chi_{3107}(250,\cdot)\) \(\chi_{3107}(256,\cdot)\) \(\chi_{3107}(269,\cdot)\) \(\chi_{3107}(289,\cdot)\) \(\chi_{3107}(341,\cdot)\) \(\chi_{3107}(347,\cdot)\) \(\chi_{3107}(360,\cdot)\) \(\chi_{3107}(373,\cdot)\) \(\chi_{3107}(386,\cdot)\) \(\chi_{3107}(399,\cdot)\) \(\chi_{3107}(419,\cdot)\) \(\chi_{3107}(425,\cdot)\) \(\chi_{3107}(432,\cdot)\) ...

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: $\Q(\zeta_{357})$
Fixed field: Number field defined by a degree 357 polynomial (not computed)

Values on generators

\((1913,963)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{45}{119}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 3107 }(425, a) \) \(1\)\(1\)\(e\left(\frac{223}{357}\right)\)\(e\left(\frac{232}{357}\right)\)\(e\left(\frac{89}{357}\right)\)\(e\left(\frac{22}{119}\right)\)\(e\left(\frac{14}{51}\right)\)\(e\left(\frac{254}{357}\right)\)\(e\left(\frac{104}{119}\right)\)\(e\left(\frac{107}{357}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{64}{357}\right)\)
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)
 
\( \chi_{ 3107 }(425,a) \;\) at \(\;a = \) e.g. 2