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Properties

Label 2301.415
Modulus $2301$
Conductor $767$
Order $58$
Real no
Primitive no
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(2301, base_ring=CyclotomicField(58)) M = H._module chi = DirichletCharacter(H, M([0,29,1]))
 
Copy content gp:[g,chi] = znchar(Mod(415, 2301))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2301.415");
 

Basic properties

Modulus: \(2301\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(767\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(58\)
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 \(\chi_{767}(415,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 2301.bb

\(\chi_{2301}(103,\cdot)\) \(\chi_{2301}(142,\cdot)\) \(\chi_{2301}(220,\cdot)\) \(\chi_{2301}(259,\cdot)\) \(\chi_{2301}(337,\cdot)\) \(\chi_{2301}(415,\cdot)\) \(\chi_{2301}(571,\cdot)\) \(\chi_{2301}(688,\cdot)\) \(\chi_{2301}(805,\cdot)\) \(\chi_{2301}(844,\cdot)\) \(\chi_{2301}(922,\cdot)\) \(\chi_{2301}(1000,\cdot)\) \(\chi_{2301}(1117,\cdot)\) \(\chi_{2301}(1234,\cdot)\) \(\chi_{2301}(1273,\cdot)\) \(\chi_{2301}(1312,\cdot)\) \(\chi_{2301}(1390,\cdot)\) \(\chi_{2301}(1429,\cdot)\) \(\chi_{2301}(1468,\cdot)\) \(\chi_{2301}(1507,\cdot)\) \(\chi_{2301}(1624,\cdot)\) \(\chi_{2301}(1663,\cdot)\) \(\chi_{2301}(1702,\cdot)\) \(\chi_{2301}(1741,\cdot)\) \(\chi_{2301}(1780,\cdot)\) \(\chi_{2301}(2014,\cdot)\) \(\chi_{2301}(2053,\cdot)\) \(\chi_{2301}(2248,\cdot)\)

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: $\Q(\zeta_{29})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 58 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((1535,886,2185)\) → \((1,-1,e\left(\frac{1}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\( \chi_{ 2301 }(415, a) \) \(-1\)\(1\)\(e\left(\frac{15}{29}\right)\)\(e\left(\frac{1}{29}\right)\)\(e\left(\frac{35}{58}\right)\)\(e\left(\frac{47}{58}\right)\)\(e\left(\frac{16}{29}\right)\)\(e\left(\frac{7}{58}\right)\)\(e\left(\frac{27}{29}\right)\)\(e\left(\frac{19}{58}\right)\)\(e\left(\frac{2}{29}\right)\)\(e\left(\frac{20}{29}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 2301 }(415,a) \;\) at \(\;a = \) e.g. 2