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Properties

Label 70601.644
Modulus $70601$
Conductor $70601$
Order $4152$
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(70601, base_ring=CyclotomicField(4152)) M = H._module chi = DirichletCharacter(H, M([1557,3526]))
 
Copy content gp:[g,chi] = znchar(Mod(644, 70601))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("70601.644");
 

Basic properties

Modulus: \(70601\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(70601\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(4152\)
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 70601.eb

\(\chi_{70601}(25,\cdot)\) \(\chi_{70601}(43,\cdot)\) \(\chi_{70601}(83,\cdot)\) \(\chi_{70601}(111,\cdot)\) \(\chi_{70601}(168,\cdot)\) \(\chi_{70601}(195,\cdot)\) \(\chi_{70601}(314,\cdot)\) \(\chi_{70601}(400,\cdot)\) \(\chi_{70601}(542,\cdot)\) \(\chi_{70601}(569,\cdot)\) \(\chi_{70601}(593,\cdot)\) \(\chi_{70601}(644,\cdot)\) \(\chi_{70601}(688,\cdot)\) \(\chi_{70601}(706,\cdot)\) \(\chi_{70601}(739,\cdot)\) \(\chi_{70601}(756,\cdot)\) \(\chi_{70601}(791,\cdot)\) \(\chi_{70601}(835,\cdot)\) \(\chi_{70601}(926,\cdot)\) \(\chi_{70601}(927,\cdot)\) \(\chi_{70601}(1001,\cdot)\) \(\chi_{70601}(1022,\cdot)\) \(\chi_{70601}(1073,\cdot)\) \(\chi_{70601}(1154,\cdot)\) \(\chi_{70601}(1164,\cdot)\) \(\chi_{70601}(1165,\cdot)\) \(\chi_{70601}(1192,\cdot)\) \(\chi_{70601}(1199,\cdot)\) \(\chi_{70601}(1328,\cdot)\) \(\chi_{70601}(1464,\cdot)\) ...

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: $\Q(\zeta_{4152})$
Fixed field: Number field defined by a degree 4152 polynomial (not computed)

Values on generators

\((58143,24923)\) → \((e\left(\frac{3}{8}\right),e\left(\frac{1763}{2076}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 70601 }(644, a) \) \(1\)\(1\)\(e\left(\frac{333}{692}\right)\)\(e\left(\frac{3877}{4152}\right)\)\(e\left(\frac{333}{346}\right)\)\(e\left(\frac{3007}{4152}\right)\)\(e\left(\frac{1723}{4152}\right)\)\(e\left(\frac{3811}{4152}\right)\)\(e\left(\frac{307}{692}\right)\)\(e\left(\frac{1801}{2076}\right)\)\(e\left(\frac{853}{4152}\right)\)\(e\left(\frac{1071}{1384}\right)\)
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)
 
\( \chi_{ 70601 }(644,a) \;\) at \(\;a = \) e.g. 2