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Properties

Label 256025.137329
Modulus $256025$
Conductor $36575$
Order $90$
Real no
Primitive no
Minimal no
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(256025, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([9,15,36,20]))
 
Copy content gp:[g,chi] = znchar(Mod(137329, 256025))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("256025.137329");
 

Basic properties

Modulus: \(256025\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(36575\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(90\)
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 \(\chi_{36575}(27604,\cdot)\)
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: no
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 256025.cce

\(\chi_{256025}(6009,\cdot)\) \(\chi_{256025}(6989,\cdot)\) \(\chi_{256025}(9194,\cdot)\) \(\chi_{256025}(10554,\cdot)\) \(\chi_{256025}(27459,\cdot)\) \(\chi_{256025}(33939,\cdot)\) \(\chi_{256025}(46434,\cdot)\) \(\chi_{256025}(55389,\cdot)\) \(\chi_{256025}(64454,\cdot)\) \(\chi_{256025}(71069,\cdot)\) \(\chi_{256025}(74364,\cdot)\) \(\chi_{256025}(108309,\cdot)\) \(\chi_{256025}(124969,\cdot)\) \(\chi_{256025}(136239,\cdot)\) \(\chi_{256025}(137329,\cdot)\) \(\chi_{256025}(162209,\cdot)\) \(\chi_{256025}(164279,\cdot)\) \(\chi_{256025}(185729,\cdot)\) \(\chi_{256025}(190139,\cdot)\) \(\chi_{256025}(197844,\cdot)\) \(\chi_{256025}(204704,\cdot)\) \(\chi_{256025}(224794,\cdot)\) \(\chi_{256025}(235084,\cdot)\) \(\chi_{256025}(246244,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((112652,198551,232751,67376)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right),e\left(\frac{2}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 256025 }(137329, a) \) \(-1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{90}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{13}{45}\right)\)
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)
 
\( \chi_{ 256025 }(137329,a) \;\) at \(\;a = \) e.g. 2