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Properties

Label 256025.42293
Modulus $256025$
Conductor $51205$
Order $140$
Real no
Primitive no
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(256025, base_ring=CyclotomicField(140)) M = H._module chi = DirichletCharacter(H, M([105,90,84,70]))
 
Copy content gp:[g,chi] = znchar(Mod(42293, 256025))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("256025.42293");
 

Basic properties

Modulus: \(256025\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(51205\)
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: no, induced from \(\chi_{51205}(42293,\cdot)\)
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 256025.cfq

\(\chi_{256025}(3457,\cdot)\) \(\chi_{256025}(5718,\cdot)\) \(\chi_{256025}(10107,\cdot)\) \(\chi_{256025}(12368,\cdot)\) \(\chi_{256025}(25668,\cdot)\) \(\chi_{256025}(30057,\cdot)\) \(\chi_{256025}(35643,\cdot)\) \(\chi_{256025}(42293,\cdot)\) \(\chi_{256025}(46682,\cdot)\) \(\chi_{256025}(48943,\cdot)\) \(\chi_{256025}(53332,\cdot)\) \(\chi_{256025}(62243,\cdot)\) \(\chi_{256025}(66632,\cdot)\) \(\chi_{256025}(72218,\cdot)\) \(\chi_{256025}(76607,\cdot)\) \(\chi_{256025}(78868,\cdot)\) \(\chi_{256025}(83257,\cdot)\) \(\chi_{256025}(85518,\cdot)\) \(\chi_{256025}(89907,\cdot)\) \(\chi_{256025}(98818,\cdot)\) \(\chi_{256025}(103207,\cdot)\) \(\chi_{256025}(108793,\cdot)\) \(\chi_{256025}(113182,\cdot)\) \(\chi_{256025}(119832,\cdot)\) \(\chi_{256025}(122093,\cdot)\) \(\chi_{256025}(126482,\cdot)\) \(\chi_{256025}(135393,\cdot)\) \(\chi_{256025}(139782,\cdot)\) \(\chi_{256025}(145368,\cdot)\) \(\chi_{256025}(149757,\cdot)\) ...

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

\((112652,198551,232751,67376)\) → \((-i,e\left(\frac{9}{14}\right),e\left(\frac{3}{5}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 256025 }(42293, a) \) \(-1\)\(1\)\(e\left(\frac{79}{140}\right)\)\(e\left(\frac{27}{140}\right)\)\(e\left(\frac{9}{70}\right)\)\(e\left(\frac{53}{70}\right)\)\(e\left(\frac{97}{140}\right)\)\(e\left(\frac{27}{70}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{79}{140}\right)\)\(e\left(\frac{9}{35}\right)\)\(e\left(\frac{31}{140}\right)\)
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)
 
\( \chi_{ 256025 }(42293,a) \;\) at \(\;a = \) e.g. 2