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Properties

Label 3664.245
Modulus $3664$
Conductor $3664$
Order $76$
Real no
Primitive yes
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(3664, base_ring=CyclotomicField(76)) M = H._module chi = DirichletCharacter(H, M([0,19,28]))
 
Copy content gp:[g,chi] = znchar(Mod(245, 3664))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3664.245");
 

Basic properties

Modulus: \(3664\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3664\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(76\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 3664.cf

\(\chi_{3664}(53,\cdot)\) \(\chi_{3664}(61,\cdot)\) \(\chi_{3664}(165,\cdot)\) \(\chi_{3664}(245,\cdot)\) \(\chi_{3664}(333,\cdot)\) \(\chi_{3664}(485,\cdot)\) \(\chi_{3664}(501,\cdot)\) \(\chi_{3664}(661,\cdot)\) \(\chi_{3664}(901,\cdot)\) \(\chi_{3664}(933,\cdot)\) \(\chi_{3664}(973,\cdot)\) \(\chi_{3664}(1037,\cdot)\) \(\chi_{3664}(1077,\cdot)\) \(\chi_{3664}(1141,\cdot)\) \(\chi_{3664}(1189,\cdot)\) \(\chi_{3664}(1205,\cdot)\) \(\chi_{3664}(1645,\cdot)\) \(\chi_{3664}(1821,\cdot)\) \(\chi_{3664}(1885,\cdot)\) \(\chi_{3664}(1893,\cdot)\) \(\chi_{3664}(1997,\cdot)\) \(\chi_{3664}(2077,\cdot)\) \(\chi_{3664}(2165,\cdot)\) \(\chi_{3664}(2317,\cdot)\) \(\chi_{3664}(2333,\cdot)\) \(\chi_{3664}(2493,\cdot)\) \(\chi_{3664}(2733,\cdot)\) \(\chi_{3664}(2765,\cdot)\) \(\chi_{3664}(2805,\cdot)\) \(\chi_{3664}(2869,\cdot)\) ...

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: $\Q(\zeta_{76})$
Fixed field: Number field defined by a degree 76 polynomial

Values on generators

\((1375,917,3441)\) → \((1,i,e\left(\frac{7}{19}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 3664 }(245, a) \) \(1\)\(1\)\(e\left(\frac{29}{76}\right)\)\(e\left(\frac{27}{76}\right)\)\(e\left(\frac{35}{38}\right)\)\(e\left(\frac{29}{38}\right)\)\(e\left(\frac{71}{76}\right)\)\(e\left(\frac{65}{76}\right)\)\(e\left(\frac{14}{19}\right)\)\(e\left(\frac{2}{19}\right)\)\(e\left(\frac{49}{76}\right)\)\(e\left(\frac{23}{76}\right)\)
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)
 
\( \chi_{ 3664 }(245,a) \;\) at \(\;a = \) e.g. 2