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Properties

Label 3526.1937
Modulus $3526$
Conductor $1763$
Order $70$
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(3526, base_ring=CyclotomicField(70)) M = H._module chi = DirichletCharacter(H, M([14,45]))
 
Copy content gp:[g,chi] = znchar(Mod(1937, 3526))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3526.1937");
 

Basic properties

Modulus: \(3526\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1763\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(70\)
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 \(\chi_{1763}(174,\cdot)\)
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 3526.bs

\(\chi_{3526}(51,\cdot)\) \(\chi_{3526}(223,\cdot)\) \(\chi_{3526}(297,\cdot)\) \(\chi_{3526}(303,\cdot)\) \(\chi_{3526}(469,\cdot)\) \(\chi_{3526}(543,\cdot)\) \(\chi_{3526}(715,\cdot)\) \(\chi_{3526}(1021,\cdot)\) \(\chi_{3526}(1451,\cdot)\) \(\chi_{3526}(1513,\cdot)\) \(\chi_{3526}(1527,\cdot)\) \(\chi_{3526}(1699,\cdot)\) \(\chi_{3526}(1759,\cdot)\) \(\chi_{3526}(1937,\cdot)\) \(\chi_{3526}(1943,\cdot)\) \(\chi_{3526}(2005,\cdot)\) \(\chi_{3526}(2109,\cdot)\) \(\chi_{3526}(2189,\cdot)\) \(\chi_{3526}(2435,\cdot)\) \(\chi_{3526}(2989,\cdot)\) \(\chi_{3526}(3085,\cdot)\) \(\chi_{3526}(3257,\cdot)\) \(\chi_{3526}(3399,\cdot)\) \(\chi_{3526}(3419,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((2753,1723)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{9}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 3526 }(1937, a) \) \(-1\)\(1\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{33}{70}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{31}{35}\right)\)\(e\left(\frac{27}{35}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{1}{35}\right)\)\(e\left(\frac{1}{70}\right)\)\(e\left(\frac{33}{35}\right)\)
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)
 
\( \chi_{ 3526 }(1937,a) \;\) at \(\;a = \) e.g. 2