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Properties

Label 8611.377
Modulus $8611$
Conductor $8611$
Order $1404$
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(8611, base_ring=CyclotomicField(1404)) M = H._module chi = DirichletCharacter(H, M([810,1313]))
 
Copy content gp:[g,chi] = znchar(Mod(377, 8611))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8611.377");
 

Basic properties

Modulus: \(8611\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(8611\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1404\)
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 8611.ez

\(\chi_{8611}(14,\cdot)\) \(\chi_{8611}(57,\cdot)\) \(\chi_{8611}(58,\cdot)\) \(\chi_{8611}(69,\cdot)\) \(\chi_{8611}(91,\cdot)\) \(\chi_{8611}(96,\cdot)\) \(\chi_{8611}(120,\cdot)\) \(\chi_{8611}(148,\cdot)\) \(\chi_{8611}(229,\cdot)\) \(\chi_{8611}(270,\cdot)\) \(\chi_{8611}(333,\cdot)\) \(\chi_{8611}(357,\cdot)\) \(\chi_{8611}(374,\cdot)\) \(\chi_{8611}(377,\cdot)\) \(\chi_{8611}(385,\cdot)\) \(\chi_{8611}(412,\cdot)\) \(\chi_{8611}(422,\cdot)\) \(\chi_{8611}(466,\cdot)\) \(\chi_{8611}(486,\cdot)\) \(\chi_{8611}(488,\cdot)\) \(\chi_{8611}(489,\cdot)\) \(\chi_{8611}(501,\cdot)\) \(\chi_{8611}(515,\cdot)\) \(\chi_{8611}(531,\cdot)\) \(\chi_{8611}(532,\cdot)\) \(\chi_{8611}(535,\cdot)\) \(\chi_{8611}(610,\cdot)\) \(\chi_{8611}(614,\cdot)\) \(\chi_{8611}(624,\cdot)\) \(\chi_{8611}(644,\cdot)\) ...

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: $\Q(\zeta_{1404})$
Fixed field: Number field defined by a degree 1404 polynomial (not computed)

Values on generators

\((6323,1423)\) → \((e\left(\frac{15}{26}\right),e\left(\frac{101}{108}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8611 }(377, a) \) \(1\)\(1\)\(e\left(\frac{287}{468}\right)\)\(e\left(\frac{145}{702}\right)\)\(e\left(\frac{53}{234}\right)\)\(e\left(\frac{296}{351}\right)\)\(e\left(\frac{1151}{1404}\right)\)\(e\left(\frac{691}{702}\right)\)\(e\left(\frac{131}{156}\right)\)\(e\left(\frac{145}{351}\right)\)\(e\left(\frac{641}{1404}\right)\)\(e\left(\frac{1195}{1404}\right)\)
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)
 
\( \chi_{ 8611 }(377,a) \;\) at \(\;a = \) e.g. 2