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Properties

Label 9652.7681
Modulus $9652$
Conductor $2413$
Order $63$
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(9652, base_ring=CyclotomicField(126)) M = H._module chi = DirichletCharacter(H, M([0,112,78]))
 
Copy content gp:[g,chi] = znchar(Mod(7681, 9652))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9652.7681");
 

Basic properties

Modulus: \(9652\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2413\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(63\)
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 \(\chi_{2413}(442,\cdot)\)
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 9652.ha

\(\chi_{9652}(25,\cdot)\) \(\chi_{9652}(177,\cdot)\) \(\chi_{9652}(301,\cdot)\) \(\chi_{9652}(481,\cdot)\) \(\chi_{9652}(625,\cdot)\) \(\chi_{9652}(757,\cdot)\) \(\chi_{9652}(1089,\cdot)\) \(\chi_{9652}(1745,\cdot)\) \(\chi_{9652}(2373,\cdot)\) \(\chi_{9652}(2601,\cdot)\) \(\chi_{9652}(2657,\cdot)\) \(\chi_{9652}(3349,\cdot)\) \(\chi_{9652}(3505,\cdot)\) \(\chi_{9652}(3581,\cdot)\) \(\chi_{9652}(3733,\cdot)\) \(\chi_{9652}(3805,\cdot)\) \(\chi_{9652}(4013,\cdot)\) \(\chi_{9652}(4037,\cdot)\) \(\chi_{9652}(4089,\cdot)\) \(\chi_{9652}(4241,\cdot)\) \(\chi_{9652}(4405,\cdot)\) \(\chi_{9652}(4545,\cdot)\) \(\chi_{9652}(4633,\cdot)\) \(\chi_{9652}(4645,\cdot)\) \(\chi_{9652}(4737,\cdot)\) \(\chi_{9652}(4793,\cdot)\) \(\chi_{9652}(5153,\cdot)\) \(\chi_{9652}(5705,\cdot)\) \(\chi_{9652}(7453,\cdot)\) \(\chi_{9652}(7681,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 63 polynomial

Values on generators

\((4827,7621,8893)\) → \((1,e\left(\frac{8}{9}\right),e\left(\frac{13}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 9652 }(7681, a) \) \(1\)\(1\)\(e\left(\frac{11}{63}\right)\)\(e\left(\frac{5}{63}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{22}{63}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{40}{63}\right)\)\(e\left(\frac{16}{63}\right)\)\(e\left(\frac{26}{63}\right)\)\(e\left(\frac{44}{63}\right)\)\(e\left(\frac{43}{63}\right)\)
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)
 
\( \chi_{ 9652 }(7681,a) \;\) at \(\;a = \) e.g. 2