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Properties

Label 5104.3175
Modulus $5104$
Conductor $2552$
Order $140$
Real no
Primitive no
Minimal no
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(5104, base_ring=CyclotomicField(140)) M = H._module chi = DirichletCharacter(H, M([70,70,98,65]))
 
Copy content gp:[g,chi] = znchar(Mod(3175, 5104))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5104.3175");
 

Basic properties

Modulus: \(5104\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2552\)
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 \(\chi_{2552}(1899,\cdot)\)
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: no
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 5104.fy

\(\chi_{5104}(39,\cdot)\) \(\chi_{5104}(327,\cdot)\) \(\chi_{5104}(359,\cdot)\) \(\chi_{5104}(391,\cdot)\) \(\chi_{5104}(503,\cdot)\) \(\chi_{5104}(519,\cdot)\) \(\chi_{5104}(711,\cdot)\) \(\chi_{5104}(743,\cdot)\) \(\chi_{5104}(855,\cdot)\) \(\chi_{5104}(1047,\cdot)\) \(\chi_{5104}(1063,\cdot)\) \(\chi_{5104}(1207,\cdot)\) \(\chi_{5104}(1239,\cdot)\) \(\chi_{5104}(1447,\cdot)\) \(\chi_{5104}(1751,\cdot)\) \(\chi_{5104}(1767,\cdot)\) \(\chi_{5104}(1911,\cdot)\) \(\chi_{5104}(1975,\cdot)\) \(\chi_{5104}(2103,\cdot)\) \(\chi_{5104}(2119,\cdot)\) \(\chi_{5104}(2439,\cdot)\) \(\chi_{5104}(2455,\cdot)\) \(\chi_{5104}(2631,\cdot)\) \(\chi_{5104}(2647,\cdot)\) \(\chi_{5104}(2679,\cdot)\) \(\chi_{5104}(2823,\cdot)\) \(\chi_{5104}(3031,\cdot)\) \(\chi_{5104}(3143,\cdot)\) \(\chi_{5104}(3159,\cdot)\) \(\chi_{5104}(3175,\cdot)\) ...

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

\((639,3829,2785,2641)\) → \((-1,-1,e\left(\frac{7}{10}\right),e\left(\frac{13}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 5104 }(3175, a) \) \(-1\)\(1\)\(e\left(\frac{129}{140}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{59}{70}\right)\)\(e\left(\frac{39}{70}\right)\)\(e\left(\frac{61}{140}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{39}{140}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{11}{14}\right)\)
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)
 
\( \chi_{ 5104 }(3175,a) \;\) at \(\;a = \) e.g. 2