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Properties

Label 3140.1379
Modulus $3140$
Conductor $3140$
Order $156$
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(3140, base_ring=CyclotomicField(156)) M = H._module chi = DirichletCharacter(H, M([78,78,103]))
 
Copy content gp:[g,chi] = znchar(Mod(1379, 3140))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3140.1379");
 

Basic properties

Modulus: \(3140\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3140\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(156\)
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 3140.cq

\(\chi_{3140}(119,\cdot)\) \(\chi_{3140}(139,\cdot)\) \(\chi_{3140}(219,\cdot)\) \(\chi_{3140}(259,\cdot)\) \(\chi_{3140}(299,\cdot)\) \(\chi_{3140}(319,\cdot)\) \(\chi_{3140}(399,\cdot)\) \(\chi_{3140}(559,\cdot)\) \(\chi_{3140}(719,\cdot)\) \(\chi_{3140}(759,\cdot)\) \(\chi_{3140}(779,\cdot)\) \(\chi_{3140}(819,\cdot)\) \(\chi_{3140}(859,\cdot)\) \(\chi_{3140}(879,\cdot)\) \(\chi_{3140}(899,\cdot)\) \(\chi_{3140}(1019,\cdot)\) \(\chi_{3140}(1039,\cdot)\) \(\chi_{3140}(1079,\cdot)\) \(\chi_{3140}(1119,\cdot)\) \(\chi_{3140}(1159,\cdot)\) \(\chi_{3140}(1179,\cdot)\) \(\chi_{3140}(1299,\cdot)\) \(\chi_{3140}(1319,\cdot)\) \(\chi_{3140}(1339,\cdot)\) \(\chi_{3140}(1379,\cdot)\) \(\chi_{3140}(1419,\cdot)\) \(\chi_{3140}(1439,\cdot)\) \(\chi_{3140}(1479,\cdot)\) \(\chi_{3140}(1639,\cdot)\) \(\chi_{3140}(1799,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((1571,1257,1261)\) → \((-1,-1,e\left(\frac{103}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 3140 }(1379, a) \) \(1\)\(1\)\(e\left(\frac{11}{78}\right)\)\(e\left(\frac{3}{52}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{77}{78}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{71}{78}\right)\)\(e\left(\frac{29}{78}\right)\)\(e\left(\frac{31}{156}\right)\)\(e\left(\frac{7}{52}\right)\)\(e\left(\frac{11}{26}\right)\)
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)
 
\( \chi_{ 3140 }(1379,a) \;\) at \(\;a = \) e.g. 2