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Properties

Label 10336.605
Modulus $10336$
Conductor $10336$
Order $144$
Real no
Primitive yes
Minimal yes
Parity odd

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(10336, base_ring=CyclotomicField(144)) M = H._module chi = DirichletCharacter(H, M([0,54,27,32]))
 
Copy content gp:[g,chi] = znchar(Mod(605, 10336))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("10336.605");
 

Basic properties

Modulus: \(10336\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(10336\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 10336.np

\(\chi_{10336}(61,\cdot)\) \(\chi_{10336}(397,\cdot)\) \(\chi_{10336}(605,\cdot)\) \(\chi_{10336}(821,\cdot)\) \(\chi_{10336}(1125,\cdot)\) \(\chi_{10336}(1149,\cdot)\) \(\chi_{10336}(1221,\cdot)\) \(\chi_{10336}(1365,\cdot)\) \(\chi_{10336}(1469,\cdot)\) \(\chi_{10336}(1525,\cdot)\) \(\chi_{10336}(1669,\cdot)\) \(\chi_{10336}(1677,\cdot)\) \(\chi_{10336}(1765,\cdot)\) \(\chi_{10336}(1909,\cdot)\) \(\chi_{10336}(2069,\cdot)\) \(\chi_{10336}(2213,\cdot)\) \(\chi_{10336}(2221,\cdot)\) \(\chi_{10336}(3101,\cdot)\) \(\chi_{10336}(3645,\cdot)\) \(\chi_{10336}(4189,\cdot)\) \(\chi_{10336}(4205,\cdot)\) \(\chi_{10336}(4413,\cdot)\) \(\chi_{10336}(4957,\cdot)\) \(\chi_{10336}(5173,\cdot)\) \(\chi_{10336}(5477,\cdot)\) \(\chi_{10336}(5573,\cdot)\) \(\chi_{10336}(5717,\cdot)\) \(\chi_{10336}(5837,\cdot)\) \(\chi_{10336}(5877,\cdot)\) \(\chi_{10336}(6021,\cdot)\) ...

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

\((3231,3877,4865,4353)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{3}{16}\right),e\left(\frac{2}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(21\)\(23\)\(25\)
\( \chi_{ 10336 }(605, a) \) \(-1\)\(1\)\(e\left(\frac{29}{144}\right)\)\(e\left(\frac{125}{144}\right)\)\(e\left(\frac{7}{48}\right)\)\(e\left(\frac{29}{72}\right)\)\(e\left(\frac{41}{48}\right)\)\(e\left(\frac{35}{72}\right)\)\(e\left(\frac{5}{72}\right)\)\(e\left(\frac{25}{72}\right)\)\(e\left(\frac{73}{144}\right)\)\(e\left(\frac{53}{72}\right)\)
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)
 
\( \chi_{ 10336 }(605,a) \;\) at \(\;a = \) e.g. 2