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Properties

Label 1405.359
Modulus $1405$
Conductor $1405$
Order $140$
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(1405, base_ring=CyclotomicField(140)) M = H._module chi = DirichletCharacter(H, M([70,107]))
 
Copy content gp:[g,chi] = znchar(Mod(359, 1405))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1405.359");
 

Basic properties

Modulus: \(1405\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1405\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(140\)
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 1405.bu

\(\chi_{1405}(9,\cdot)\) \(\chi_{1405}(14,\cdot)\) \(\chi_{1405}(69,\cdot)\) \(\chi_{1405}(114,\cdot)\) \(\chi_{1405}(144,\cdot)\) \(\chi_{1405}(149,\cdot)\) \(\chi_{1405}(169,\cdot)\) \(\chi_{1405}(209,\cdot)\) \(\chi_{1405}(224,\cdot)\) \(\chi_{1405}(264,\cdot)\) \(\chi_{1405}(299,\cdot)\) \(\chi_{1405}(314,\cdot)\) \(\chi_{1405}(359,\cdot)\) \(\chi_{1405}(419,\cdot)\) \(\chi_{1405}(424,\cdot)\) \(\chi_{1405}(484,\cdot)\) \(\chi_{1405}(529,\cdot)\) \(\chi_{1405}(544,\cdot)\) \(\chi_{1405}(579,\cdot)\) \(\chi_{1405}(619,\cdot)\) \(\chi_{1405}(634,\cdot)\) \(\chi_{1405}(674,\cdot)\) \(\chi_{1405}(694,\cdot)\) \(\chi_{1405}(699,\cdot)\) \(\chi_{1405}(729,\cdot)\) \(\chi_{1405}(774,\cdot)\) \(\chi_{1405}(829,\cdot)\) \(\chi_{1405}(834,\cdot)\) \(\chi_{1405}(874,\cdot)\) \(\chi_{1405}(899,\cdot)\) ...

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: $\Q(\zeta_{140})$
Fixed field: Number field defined by a degree 140 polynomial (not computed)

Values on generators

\((282,846)\) → \((-1,e\left(\frac{107}{140}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1405 }(359, a) \) \(1\)\(1\)\(e\left(\frac{29}{70}\right)\)\(e\left(\frac{37}{140}\right)\)\(e\left(\frac{29}{35}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{17}{70}\right)\)\(e\left(\frac{37}{70}\right)\)\(e\left(\frac{51}{140}\right)\)\(e\left(\frac{13}{140}\right)\)\(e\left(\frac{53}{140}\right)\)
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)
 
\( \chi_{ 1405 }(359,a) \;\) at \(\;a = \) e.g. 2