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Properties

Label 32704.18141
Modulus $32704$
Conductor $32704$
Order $144$
Real no
Primitive yes
Minimal yes
Parity even

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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(32704, base_ring=CyclotomicField(144)) M = H._module chi = DirichletCharacter(H, M([0,99,96,128]))
 
Copy content gp:[g,chi] = znchar(Mod(18141, 32704))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("32704.18141");
 

Basic properties

Modulus: \(32704\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(32704\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(144\)
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 32704.bis

\(\chi_{32704}(1549,\cdot)\) \(\chi_{32704}(1565,\cdot)\) \(\chi_{32704}(1661,\cdot)\) \(\chi_{32704}(1789,\cdot)\) \(\chi_{32704}(2557,\cdot)\) \(\chi_{32704}(3581,\cdot)\) \(\chi_{32704}(5637,\cdot)\) \(\chi_{32704}(5653,\cdot)\) \(\chi_{32704}(5749,\cdot)\) \(\chi_{32704}(5877,\cdot)\) \(\chi_{32704}(6645,\cdot)\) \(\chi_{32704}(7669,\cdot)\) \(\chi_{32704}(9725,\cdot)\) \(\chi_{32704}(9741,\cdot)\) \(\chi_{32704}(9837,\cdot)\) \(\chi_{32704}(9965,\cdot)\) \(\chi_{32704}(10733,\cdot)\) \(\chi_{32704}(11757,\cdot)\) \(\chi_{32704}(13813,\cdot)\) \(\chi_{32704}(13829,\cdot)\) \(\chi_{32704}(13925,\cdot)\) \(\chi_{32704}(14053,\cdot)\) \(\chi_{32704}(14821,\cdot)\) \(\chi_{32704}(15845,\cdot)\) \(\chi_{32704}(17901,\cdot)\) \(\chi_{32704}(17917,\cdot)\) \(\chi_{32704}(18013,\cdot)\) \(\chi_{32704}(18141,\cdot)\) \(\chi_{32704}(18909,\cdot)\) \(\chi_{32704}(19933,\cdot)\) ...

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: $\Q(\zeta_{144})$
Fixed field: Number field defined by a degree 144 polynomial (not computed)

Values on generators

\((1023,30661,14017,6721)\) → \((1,e\left(\frac{11}{16}\right),e\left(\frac{2}{3}\right),e\left(\frac{8}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 32704 }(18141, a) \) \(1\)\(1\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{131}{144}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{143}{144}\right)\)\(e\left(\frac{109}{144}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{37}{144}\right)\)\(e\left(\frac{61}{72}\right)\)\(e\left(\frac{59}{72}\right)\)
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)
 
\( \chi_{ 32704 }(18141,a) \;\) at \(\;a = \) e.g. 2