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Properties

Label 4025.1019
Modulus $4025$
Conductor $4025$
Order $330$
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(4025, base_ring=CyclotomicField(330)) M = H._module chi = DirichletCharacter(H, M([297,220,285]))
 
Copy content gp:[g,chi] = znchar(Mod(1019, 4025))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4025.1019");
 

Basic properties

Modulus: \(4025\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4025\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(330\)
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 4025.dn

\(\chi_{4025}(44,\cdot)\) \(\chi_{4025}(79,\cdot)\) \(\chi_{4025}(109,\cdot)\) \(\chi_{4025}(214,\cdot)\) \(\chi_{4025}(319,\cdot)\) \(\chi_{4025}(359,\cdot)\) \(\chi_{4025}(389,\cdot)\) \(\chi_{4025}(429,\cdot)\) \(\chi_{4025}(494,\cdot)\) \(\chi_{4025}(534,\cdot)\) \(\chi_{4025}(569,\cdot)\) \(\chi_{4025}(704,\cdot)\) \(\chi_{4025}(709,\cdot)\) \(\chi_{4025}(779,\cdot)\) \(\chi_{4025}(879,\cdot)\) \(\chi_{4025}(884,\cdot)\) \(\chi_{4025}(914,\cdot)\) \(\chi_{4025}(954,\cdot)\) \(\chi_{4025}(1019,\cdot)\) \(\chi_{4025}(1054,\cdot)\) \(\chi_{4025}(1164,\cdot)\) \(\chi_{4025}(1194,\cdot)\) \(\chi_{4025}(1229,\cdot)\) \(\chi_{4025}(1234,\cdot)\) \(\chi_{4025}(1339,\cdot)\) \(\chi_{4025}(1479,\cdot)\) \(\chi_{4025}(1509,\cdot)\) \(\chi_{4025}(1514,\cdot)\) \(\chi_{4025}(1579,\cdot)\) \(\chi_{4025}(1584,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((2577,1151,3501)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{2}{3}\right),e\left(\frac{19}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(1019, a) \) \(-1\)\(1\)\(e\left(\frac{317}{330}\right)\)\(e\left(\frac{259}{330}\right)\)\(e\left(\frac{152}{165}\right)\)\(e\left(\frac{41}{55}\right)\)\(e\left(\frac{97}{110}\right)\)\(e\left(\frac{94}{165}\right)\)\(e\left(\frac{277}{330}\right)\)\(e\left(\frac{233}{330}\right)\)\(e\left(\frac{21}{110}\right)\)\(e\left(\frac{139}{165}\right)\)
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)
 
\( \chi_{ 4025 }(1019,a) \;\) at \(\;a = \) e.g. 2