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Properties

Label 1207.336
Modulus $1207$
Conductor $1207$
Order $140$
Real no
Primitive yes
Minimal yes
Parity odd

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(1207, base_ring=CyclotomicField(140)) M = H._module chi = DirichletCharacter(H, M([35,102]))
 
Copy content gp:[g,chi] = znchar(Mod(336, 1207))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1207.336");
 

Basic properties

Modulus: \(1207\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1207\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1207.bi

\(\chi_{1207}(13,\cdot)\) \(\chi_{1207}(21,\cdot)\) \(\chi_{1207}(47,\cdot)\) \(\chi_{1207}(55,\cdot)\) \(\chi_{1207}(106,\cdot)\) \(\chi_{1207}(115,\cdot)\) \(\chi_{1207}(123,\cdot)\) \(\chi_{1207}(132,\cdot)\) \(\chi_{1207}(140,\cdot)\) \(\chi_{1207}(149,\cdot)\) \(\chi_{1207}(234,\cdot)\) \(\chi_{1207}(268,\cdot)\) \(\chi_{1207}(276,\cdot)\) \(\chi_{1207}(319,\cdot)\) \(\chi_{1207}(336,\cdot)\) \(\chi_{1207}(353,\cdot)\) \(\chi_{1207}(489,\cdot)\) \(\chi_{1207}(565,\cdot)\) \(\chi_{1207}(599,\cdot)\) \(\chi_{1207}(633,\cdot)\) \(\chi_{1207}(650,\cdot)\) \(\chi_{1207}(667,\cdot)\) \(\chi_{1207}(701,\cdot)\) \(\chi_{1207}(752,\cdot)\) \(\chi_{1207}(769,\cdot)\) \(\chi_{1207}(778,\cdot)\) \(\chi_{1207}(803,\cdot)\) \(\chi_{1207}(812,\cdot)\) \(\chi_{1207}(837,\cdot)\) \(\chi_{1207}(846,\cdot)\) ...

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

\((853,1072)\) → \((i,e\left(\frac{51}{70}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1207 }(336, a) \) \(-1\)\(1\)\(e\left(\frac{61}{70}\right)\)\(e\left(\frac{27}{140}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{9}{140}\right)\)\(e\left(\frac{67}{140}\right)\)\(e\left(\frac{43}{70}\right)\)\(e\left(\frac{27}{70}\right)\)\(e\left(\frac{73}{140}\right)\)\(e\left(\frac{47}{140}\right)\)
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)
 
\( \chi_{ 1207 }(336,a) \;\) at \(\;a = \) e.g. 2