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Properties

Label 1203.101
Modulus $1203$
Conductor $1203$
Order $400$
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(1203, base_ring=CyclotomicField(400)) M = H._module chi = DirichletCharacter(H, M([200,349]))
 
Copy content gp:[g,chi] = znchar(Mod(101, 1203))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1203.101");
 

Basic properties

Modulus: \(1203\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1203\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(400\)
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 1203.bc

\(\chi_{1203}(17,\cdot)\) \(\chi_{1203}(23,\cdot)\) \(\chi_{1203}(38,\cdot)\) \(\chi_{1203}(53,\cdot)\) \(\chi_{1203}(59,\cdot)\) \(\chi_{1203}(62,\cdot)\) \(\chi_{1203}(65,\cdot)\) \(\chi_{1203}(71,\cdot)\) \(\chi_{1203}(74,\cdot)\) \(\chi_{1203}(92,\cdot)\) \(\chi_{1203}(95,\cdot)\) \(\chi_{1203}(101,\cdot)\) \(\chi_{1203}(104,\cdot)\) \(\chi_{1203}(107,\cdot)\) \(\chi_{1203}(122,\cdot)\) \(\chi_{1203}(131,\cdot)\) \(\chi_{1203}(134,\cdot)\) \(\chi_{1203}(137,\cdot)\) \(\chi_{1203}(143,\cdot)\) \(\chi_{1203}(152,\cdot)\) \(\chi_{1203}(161,\cdot)\) \(\chi_{1203}(167,\cdot)\) \(\chi_{1203}(170,\cdot)\) \(\chi_{1203}(182,\cdot)\) \(\chi_{1203}(185,\cdot)\) \(\chi_{1203}(191,\cdot)\) \(\chi_{1203}(194,\cdot)\) \(\chi_{1203}(209,\cdot)\) \(\chi_{1203}(227,\cdot)\) \(\chi_{1203}(233,\cdot)\) ...

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: $\Q(\zeta_{400})$
Fixed field: Number field defined by a degree 400 polynomial (not computed)

Values on generators

\((803,805)\) → \((-1,e\left(\frac{349}{400}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1203 }(101, a) \) \(1\)\(1\)\(e\left(\frac{37}{200}\right)\)\(e\left(\frac{37}{100}\right)\)\(e\left(\frac{19}{50}\right)\)\(e\left(\frac{19}{200}\right)\)\(e\left(\frac{111}{200}\right)\)\(e\left(\frac{113}{200}\right)\)\(e\left(\frac{153}{200}\right)\)\(e\left(\frac{131}{400}\right)\)\(e\left(\frac{7}{25}\right)\)\(e\left(\frac{37}{50}\right)\)
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)
 
\( \chi_{ 1203 }(101,a) \;\) at \(\;a = \) e.g. 2