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Properties

Label 2025.367
Modulus $2025$
Conductor $2025$
Order $540$
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(2025, base_ring=CyclotomicField(540)) M = H._module chi = DirichletCharacter(H, M([220,351]))
 
Copy content gp:[g,chi] = znchar(Mod(367, 2025))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2025.367");
 

Basic properties

Modulus: \(2025\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2025\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(540\)
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 2025.bu

\(\chi_{2025}(13,\cdot)\) \(\chi_{2025}(22,\cdot)\) \(\chi_{2025}(52,\cdot)\) \(\chi_{2025}(58,\cdot)\) \(\chi_{2025}(67,\cdot)\) \(\chi_{2025}(88,\cdot)\) \(\chi_{2025}(97,\cdot)\) \(\chi_{2025}(103,\cdot)\) \(\chi_{2025}(112,\cdot)\) \(\chi_{2025}(133,\cdot)\) \(\chi_{2025}(142,\cdot)\) \(\chi_{2025}(148,\cdot)\) \(\chi_{2025}(178,\cdot)\) \(\chi_{2025}(187,\cdot)\) \(\chi_{2025}(202,\cdot)\) \(\chi_{2025}(223,\cdot)\) \(\chi_{2025}(238,\cdot)\) \(\chi_{2025}(247,\cdot)\) \(\chi_{2025}(277,\cdot)\) \(\chi_{2025}(283,\cdot)\) \(\chi_{2025}(292,\cdot)\) \(\chi_{2025}(313,\cdot)\) \(\chi_{2025}(322,\cdot)\) \(\chi_{2025}(328,\cdot)\) \(\chi_{2025}(337,\cdot)\) \(\chi_{2025}(358,\cdot)\) \(\chi_{2025}(367,\cdot)\) \(\chi_{2025}(373,\cdot)\) \(\chi_{2025}(403,\cdot)\) \(\chi_{2025}(412,\cdot)\) ...

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: $\Q(\zeta_{540})$
Fixed field: Number field defined by a degree 540 polynomial (not computed)

Values on generators

\((326,1702)\) → \((e\left(\frac{11}{27}\right),e\left(\frac{13}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 2025 }(367, a) \) \(-1\)\(1\)\(e\left(\frac{31}{540}\right)\)\(e\left(\frac{31}{270}\right)\)\(e\left(\frac{83}{108}\right)\)\(e\left(\frac{31}{180}\right)\)\(e\left(\frac{94}{135}\right)\)\(e\left(\frac{329}{540}\right)\)\(e\left(\frac{223}{270}\right)\)\(e\left(\frac{31}{135}\right)\)\(e\left(\frac{161}{180}\right)\)\(e\left(\frac{23}{90}\right)\)
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)
 
\( \chi_{ 2025 }(367,a) \;\) at \(\;a = \) e.g. 2