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Properties

Label 37884.3889
Modulus $37884$
Conductor $3157$
Order $120$
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(37884, base_ring=CyclotomicField(120)) M = H._module chi = DirichletCharacter(H, M([0,0,80,108,63]))
 
Copy content gp:[g,chi] = znchar(Mod(3889, 37884))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("37884.3889");
 

Basic properties

Modulus: \(37884\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3157\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(120\)
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 \(\chi_{3157}(732,\cdot)\)
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 37884.bfq

\(\chi_{37884}(457,\cdot)\) \(\chi_{37884}(1381,\cdot)\) \(\chi_{37884}(2125,\cdot)\) \(\chi_{37884}(2965,\cdot)\) \(\chi_{37884}(3889,\cdot)\) \(\chi_{37884}(5233,\cdot)\) \(\chi_{37884}(7081,\cdot)\) \(\chi_{37884}(11041,\cdot)\) \(\chi_{37884}(11281,\cdot)\) \(\chi_{37884}(12205,\cdot)\) \(\chi_{37884}(12553,\cdot)\) \(\chi_{37884}(12889,\cdot)\) \(\chi_{37884}(14989,\cdot)\) \(\chi_{37884}(18685,\cdot)\) \(\chi_{37884}(19609,\cdot)\) \(\chi_{37884}(19945,\cdot)\) \(\chi_{37884}(21793,\cdot)\) \(\chi_{37884}(21865,\cdot)\) \(\chi_{37884}(23305,\cdot)\) \(\chi_{37884}(23377,\cdot)\) \(\chi_{37884}(23713,\cdot)\) \(\chi_{37884}(25813,\cdot)\) \(\chi_{37884}(29185,\cdot)\) \(\chi_{37884}(29509,\cdot)\) \(\chi_{37884}(30025,\cdot)\) \(\chi_{37884}(30433,\cdot)\) \(\chi_{37884}(30769,\cdot)\) \(\chi_{37884}(30949,\cdot)\) \(\chi_{37884}(32293,\cdot)\) \(\chi_{37884}(32617,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((18943,12629,10825,3445,15709)\) → \((1,1,e\left(\frac{2}{3}\right),e\left(\frac{9}{10}\right),e\left(\frac{21}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(43\)
\( \chi_{ 37884 }(3889, a) \) \(1\)\(1\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{11}{120}\right)\)\(e\left(\frac{91}{120}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{39}{40}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{3}{20}\right)\)
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)
 
\( \chi_{ 37884 }(3889,a) \;\) at \(\;a = \) e.g. 2