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Properties

Label 2552.1907
Modulus $2552$
Conductor $2552$
Order $70$
Real no
Primitive yes
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(2552, base_ring=CyclotomicField(70)) M = H._module chi = DirichletCharacter(H, M([35,35,14,65]))
 
Copy content gp:[g,chi] = znchar(Mod(1907, 2552))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2552.1907");
 

Basic properties

Modulus: \(2552\)
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: \(70\)
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: 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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 2552.cv

\(\chi_{2552}(91,\cdot)\) \(\chi_{2552}(179,\cdot)\) \(\chi_{2552}(267,\cdot)\) \(\chi_{2552}(323,\cdot)\) \(\chi_{2552}(411,\cdot)\) \(\chi_{2552}(499,\cdot)\) \(\chi_{2552}(515,\cdot)\) \(\chi_{2552}(531,\cdot)\) \(\chi_{2552}(555,\cdot)\) \(\chi_{2552}(643,\cdot)\) \(\chi_{2552}(731,\cdot)\) \(\chi_{2552}(763,\cdot)\) \(\chi_{2552}(883,\cdot)\) \(\chi_{2552}(995,\cdot)\) \(\chi_{2552}(1115,\cdot)\) \(\chi_{2552}(1347,\cdot)\) \(\chi_{2552}(1483,\cdot)\) \(\chi_{2552}(1571,\cdot)\) \(\chi_{2552}(1659,\cdot)\) \(\chi_{2552}(1675,\cdot)\) \(\chi_{2552}(1907,\cdot)\) \(\chi_{2552}(1923,\cdot)\) \(\chi_{2552}(2139,\cdot)\) \(\chi_{2552}(2275,\cdot)\)

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

\((639,1277,233,89)\) → \((-1,-1,e\left(\frac{1}{5}\right),e\left(\frac{13}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 2552 }(1907, a) \) \(-1\)\(1\)\(e\left(\frac{17}{70}\right)\)\(e\left(\frac{51}{70}\right)\)\(e\left(\frac{3}{70}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{29}{70}\right)\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{67}{70}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{1}{14}\right)\)
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)
 
\( \chi_{ 2552 }(1907,a) \;\) at \(\;a = \) e.g. 2