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Properties

Label 7975.3593
Modulus $7975$
Conductor $1595$
Order $140$
Real no
Primitive no
Minimal yes
Parity odd

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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(7975, base_ring=CyclotomicField(140)) M = H._module chi = DirichletCharacter(H, M([105,98,95]))
 
Copy content gp:[g,chi] = znchar(Mod(3593, 7975))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7975.3593");
 

Basic properties

Modulus: \(7975\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1595\)
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 \(\chi_{1595}(403,\cdot)\)
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 7975.ie

\(\chi_{7975}(18,\cdot)\) \(\chi_{7975}(68,\cdot)\) \(\chi_{7975}(282,\cdot)\) \(\chi_{7975}(657,\cdot)\) \(\chi_{7975}(743,\cdot)\) \(\chi_{7975}(943,\cdot)\) \(\chi_{7975}(1007,\cdot)\) \(\chi_{7975}(1168,\cdot)\) \(\chi_{7975}(1382,\cdot)\) \(\chi_{7975}(1432,\cdot)\) \(\chi_{7975}(1482,\cdot)\) \(\chi_{7975}(1668,\cdot)\) \(\chi_{7975}(2032,\cdot)\) \(\chi_{7975}(2107,\cdot)\) \(\chi_{7975}(2207,\cdot)\) \(\chi_{7975}(2318,\cdot)\) \(\chi_{7975}(2393,\cdot)\) \(\chi_{7975}(2757,\cdot)\) \(\chi_{7975}(2868,\cdot)\) \(\chi_{7975}(2932,\cdot)\) \(\chi_{7975}(3043,\cdot)\) \(\chi_{7975}(3407,\cdot)\) \(\chi_{7975}(3482,\cdot)\) \(\chi_{7975}(3593,\cdot)\) \(\chi_{7975}(3643,\cdot)\) \(\chi_{7975}(3693,\cdot)\) \(\chi_{7975}(3768,\cdot)\) \(\chi_{7975}(3907,\cdot)\) \(\chi_{7975}(4132,\cdot)\) \(\chi_{7975}(4318,\cdot)\) ...

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

\((1277,7251,7426)\) → \((-i,e\left(\frac{7}{10}\right),e\left(\frac{19}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 7975 }(3593, a) \) \(-1\)\(1\)\(e\left(\frac{9}{70}\right)\)\(e\left(\frac{17}{70}\right)\)\(e\left(\frac{9}{35}\right)\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{111}{140}\right)\)\(e\left(\frac{27}{70}\right)\)\(e\left(\frac{17}{35}\right)\)\(-1\)\(e\left(\frac{23}{140}\right)\)\(e\left(\frac{129}{140}\right)\)
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)
 
\( \chi_{ 7975 }(3593,a) \;\) at \(\;a = \) e.g. 2