↑

Properties

Label 2217.182
Modulus $2217$
Conductor $2217$
Order $738$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2217, base_ring=CyclotomicField(738)) M = H._module chi = DirichletCharacter(H, M([369,565]))
 
Copy content gp:[g,chi] = znchar(Mod(182, 2217))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2217.182");
 

Basic properties

Modulus: \(2217\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2217\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(738\)
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 2217.x

\(\chi_{2217}(23,\cdot)\) \(\chi_{2217}(29,\cdot)\) \(\chi_{2217}(35,\cdot)\) \(\chi_{2217}(62,\cdot)\) \(\chi_{2217}(71,\cdot)\) \(\chi_{2217}(77,\cdot)\) \(\chi_{2217}(83,\cdot)\) \(\chi_{2217}(86,\cdot)\) \(\chi_{2217}(92,\cdot)\) \(\chi_{2217}(95,\cdot)\) \(\chi_{2217}(98,\cdot)\) \(\chi_{2217}(101,\cdot)\) \(\chi_{2217}(110,\cdot)\) \(\chi_{2217}(116,\cdot)\) \(\chi_{2217}(119,\cdot)\) \(\chi_{2217}(131,\cdot)\) \(\chi_{2217}(134,\cdot)\) \(\chi_{2217}(140,\cdot)\) \(\chi_{2217}(143,\cdot)\) \(\chi_{2217}(146,\cdot)\) \(\chi_{2217}(158,\cdot)\) \(\chi_{2217}(170,\cdot)\) \(\chi_{2217}(182,\cdot)\) \(\chi_{2217}(221,\cdot)\) \(\chi_{2217}(233,\cdot)\) \(\chi_{2217}(242,\cdot)\) \(\chi_{2217}(248,\cdot)\) \(\chi_{2217}(281,\cdot)\) \(\chi_{2217}(284,\cdot)\) \(\chi_{2217}(287,\cdot)\) ...

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: $\Q(\zeta_{369})$
Fixed field: Number field defined by a degree 738 polynomial (not computed)

Values on generators

\((740,742)\) → \((-1,e\left(\frac{565}{738}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 2217 }(182, a) \) \(1\)\(1\)\(e\left(\frac{92}{123}\right)\)\(e\left(\frac{61}{123}\right)\)\(e\left(\frac{85}{246}\right)\)\(e\left(\frac{511}{738}\right)\)\(e\left(\frac{10}{41}\right)\)\(e\left(\frac{23}{246}\right)\)\(e\left(\frac{37}{738}\right)\)\(e\left(\frac{151}{246}\right)\)\(e\left(\frac{325}{738}\right)\)\(e\left(\frac{122}{123}\right)\)
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)
 
\( \chi_{ 2217 }(182,a) \;\) at \(\;a = \) e.g. 2