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Properties

Label 5924.315
Modulus $5924$
Conductor $5924$
Order $370$
Real no
Primitive yes
Minimal yes
Parity odd

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(5924, base_ring=CyclotomicField(370)) M = H._module chi = DirichletCharacter(H, M([185,66]))
 
Copy content gp:[g,chi] = znchar(Mod(315, 5924))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5924.315");
 

Basic properties

Modulus: \(5924\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5924\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(370\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 5924.z

\(\chi_{5924}(7,\cdot)\) \(\chi_{5924}(91,\cdot)\) \(\chi_{5924}(107,\cdot)\) \(\chi_{5924}(179,\cdot)\) \(\chi_{5924}(207,\cdot)\) \(\chi_{5924}(219,\cdot)\) \(\chi_{5924}(235,\cdot)\) \(\chi_{5924}(287,\cdot)\) \(\chi_{5924}(311,\cdot)\) \(\chi_{5924}(315,\cdot)\) \(\chi_{5924}(343,\cdot)\) \(\chi_{5924}(415,\cdot)\) \(\chi_{5924}(583,\cdot)\) \(\chi_{5924}(623,\cdot)\) \(\chi_{5924}(631,\cdot)\) \(\chi_{5924}(651,\cdot)\) \(\chi_{5924}(695,\cdot)\) \(\chi_{5924}(851,\cdot)\) \(\chi_{5924}(859,\cdot)\) \(\chi_{5924}(903,\cdot)\) \(\chi_{5924}(907,\cdot)\) \(\chi_{5924}(1019,\cdot)\) \(\chi_{5924}(1067,\cdot)\) \(\chi_{5924}(1111,\cdot)\) \(\chi_{5924}(1123,\cdot)\) \(\chi_{5924}(1143,\cdot)\) \(\chi_{5924}(1183,\cdot)\) \(\chi_{5924}(1223,\cdot)\) \(\chi_{5924}(1267,\cdot)\) \(\chi_{5924}(1299,\cdot)\) ...

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: $\Q(\zeta_{185})$
Fixed field: Number field defined by a degree 370 polynomial (not computed)

Values on generators

\((2963,2965)\) → \((-1,e\left(\frac{33}{185}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 5924 }(315, a) \) \(-1\)\(1\)\(e\left(\frac{251}{370}\right)\)\(e\left(\frac{79}{185}\right)\)\(e\left(\frac{299}{370}\right)\)\(e\left(\frac{66}{185}\right)\)\(e\left(\frac{67}{74}\right)\)\(e\left(\frac{30}{37}\right)\)\(e\left(\frac{39}{370}\right)\)\(e\left(\frac{24}{37}\right)\)\(e\left(\frac{121}{370}\right)\)\(e\left(\frac{18}{37}\right)\)
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)
 
\( \chi_{ 5924 }(315,a) \;\) at \(\;a = \) e.g. 2