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Properties

Label 2572.77
Modulus $2572$
Conductor $643$
Order $214$
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(2572, base_ring=CyclotomicField(214)) M = H._module chi = DirichletCharacter(H, M([0,177]))
 
Copy content gp:[g,chi] = znchar(Mod(77, 2572))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2572.77");
 

Basic properties

Modulus: \(2572\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(643\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(214\)
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 \(\chi_{643}(77,\cdot)\)
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 2572.l

\(\chi_{2572}(5,\cdot)\) \(\chi_{2572}(45,\cdot)\) \(\chi_{2572}(77,\cdot)\) \(\chi_{2572}(125,\cdot)\) \(\chi_{2572}(157,\cdot)\) \(\chi_{2572}(329,\cdot)\) \(\chi_{2572}(389,\cdot)\) \(\chi_{2572}(393,\cdot)\) \(\chi_{2572}(405,\cdot)\) \(\chi_{2572}(429,\cdot)\) \(\chi_{2572}(437,\cdot)\) \(\chi_{2572}(489,\cdot)\) \(\chi_{2572}(493,\cdot)\) \(\chi_{2572}(501,\cdot)\) \(\chi_{2572}(509,\cdot)\) \(\chi_{2572}(553,\cdot)\) \(\chi_{2572}(557,\cdot)\) \(\chi_{2572}(589,\cdot)\) \(\chi_{2572}(633,\cdot)\) \(\chi_{2572}(637,\cdot)\) \(\chi_{2572}(645,\cdot)\) \(\chi_{2572}(661,\cdot)\) \(\chi_{2572}(673,\cdot)\) \(\chi_{2572}(693,\cdot)\) \(\chi_{2572}(805,\cdot)\) \(\chi_{2572}(901,\cdot)\) \(\chi_{2572}(905,\cdot)\) \(\chi_{2572}(913,\cdot)\) \(\chi_{2572}(929,\cdot)\) \(\chi_{2572}(965,\cdot)\) ...

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: $\Q(\zeta_{107})$
Fixed field: Number field defined by a degree 214 polynomial (not computed)

Values on generators

\((1287,1297)\) → \((1,e\left(\frac{177}{214}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 2572 }(77, a) \) \(-1\)\(1\)\(e\left(\frac{123}{214}\right)\)\(e\left(\frac{71}{214}\right)\)\(e\left(\frac{39}{107}\right)\)\(e\left(\frac{16}{107}\right)\)\(e\left(\frac{177}{214}\right)\)\(e\left(\frac{5}{214}\right)\)\(e\left(\frac{97}{107}\right)\)\(e\left(\frac{19}{214}\right)\)\(e\left(\frac{173}{214}\right)\)\(e\left(\frac{201}{214}\right)\)
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)
 
\( \chi_{ 2572 }(77,a) \;\) at \(\;a = \) e.g. 2