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Properties

Label 28900.37
Modulus $28900$
Conductor $7225$
Order $1360$
Real no
Primitive no
Minimal yes
Parity even

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(28900, base_ring=CyclotomicField(1360)) M = H._module chi = DirichletCharacter(H, M([0,612,645]))
 
Copy content gp:[g,chi] = znchar(Mod(37, 28900))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("28900.37");
 

Basic properties

Modulus: \(28900\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7225\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1360\)
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 \(\chi_{7225}(37,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 28900.fm

\(\chi_{28900}(37,\cdot)\) \(\chi_{28900}(97,\cdot)\) \(\chi_{28900}(113,\cdot)\) \(\chi_{28900}(277,\cdot)\) \(\chi_{28900}(313,\cdot)\) \(\chi_{28900}(333,\cdot)\) \(\chi_{28900}(337,\cdot)\) \(\chi_{28900}(377,\cdot)\) \(\chi_{28900}(437,\cdot)\) \(\chi_{28900}(453,\cdot)\) \(\chi_{28900}(533,\cdot)\) \(\chi_{28900}(617,\cdot)\) \(\chi_{28900}(673,\cdot)\) \(\chi_{28900}(677,\cdot)\) \(\chi_{28900}(717,\cdot)\) \(\chi_{28900}(777,\cdot)\) \(\chi_{28900}(873,\cdot)\) \(\chi_{28900}(1013,\cdot)\) \(\chi_{28900}(1017,\cdot)\) \(\chi_{28900}(1117,\cdot)\) \(\chi_{28900}(1133,\cdot)\) \(\chi_{28900}(1213,\cdot)\) \(\chi_{28900}(1297,\cdot)\) \(\chi_{28900}(1333,\cdot)\) \(\chi_{28900}(1353,\cdot)\) \(\chi_{28900}(1397,\cdot)\) \(\chi_{28900}(1473,\cdot)\)

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: $\Q(\zeta_{1360})$
Fixed field: Number field defined by a degree 1360 polynomial (not computed)

Values on generators

\((14451,24277,23701)\) → \((1,e\left(\frac{9}{20}\right),e\left(\frac{129}{272}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 28900 }(37, a) \) \(1\)\(1\)\(e\left(\frac{849}{1360}\right)\)\(e\left(\frac{207}{272}\right)\)\(e\left(\frac{169}{680}\right)\)\(e\left(\frac{147}{1360}\right)\)\(e\left(\frac{43}{85}\right)\)\(e\left(\frac{503}{680}\right)\)\(e\left(\frac{131}{340}\right)\)\(e\left(\frac{967}{1360}\right)\)\(e\left(\frac{1187}{1360}\right)\)\(e\left(\frac{249}{1360}\right)\)
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)
 
\( \chi_{ 28900 }(37,a) \;\) at \(\;a = \) e.g. 2