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Properties

Label 1555.137
Modulus $1555$
Conductor $1555$
Order $620$
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(1555, base_ring=CyclotomicField(620)) M = H._module chi = DirichletCharacter(H, M([155,544]))
 
Copy content gp:[g,chi] = znchar(Mod(137, 1555))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1555.137");
 

Basic properties

Modulus: \(1555\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1555\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(620\)
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 1555.w

\(\chi_{1555}(2,\cdot)\) \(\chi_{1555}(3,\cdot)\) \(\chi_{1555}(8,\cdot)\) \(\chi_{1555}(12,\cdot)\) \(\chi_{1555}(27,\cdot)\) \(\chi_{1555}(28,\cdot)\) \(\chi_{1555}(42,\cdot)\) \(\chi_{1555}(48,\cdot)\) \(\chi_{1555}(53,\cdot)\) \(\chi_{1555}(63,\cdot)\) \(\chi_{1555}(67,\cdot)\) \(\chi_{1555}(72,\cdot)\) \(\chi_{1555}(73,\cdot)\) \(\chi_{1555}(78,\cdot)\) \(\chi_{1555}(98,\cdot)\) \(\chi_{1555}(107,\cdot)\) \(\chi_{1555}(108,\cdot)\) \(\chi_{1555}(112,\cdot)\) \(\chi_{1555}(117,\cdot)\) \(\chi_{1555}(127,\cdot)\) \(\chi_{1555}(128,\cdot)\) \(\chi_{1555}(137,\cdot)\) \(\chi_{1555}(147,\cdot)\) \(\chi_{1555}(157,\cdot)\) \(\chi_{1555}(158,\cdot)\) \(\chi_{1555}(162,\cdot)\) \(\chi_{1555}(163,\cdot)\) \(\chi_{1555}(173,\cdot)\) \(\chi_{1555}(178,\cdot)\) \(\chi_{1555}(182,\cdot)\) ...

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: $\Q(\zeta_{620})$
Fixed field: Number field defined by a degree 620 polynomial (not computed)

Values on generators

\((312,1261)\) → \((i,e\left(\frac{136}{155}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1555 }(137, a) \) \(-1\)\(1\)\(e\left(\frac{343}{620}\right)\)\(e\left(\frac{401}{620}\right)\)\(e\left(\frac{33}{310}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{11}{124}\right)\)\(e\left(\frac{409}{620}\right)\)\(e\left(\frac{91}{310}\right)\)\(e\left(\frac{14}{31}\right)\)\(e\left(\frac{467}{620}\right)\)\(e\left(\frac{117}{124}\right)\)
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)
 
\( \chi_{ 1555 }(137,a) \;\) at \(\;a = \) e.g. 2