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Properties

Label 72963.26674
Modulus $72963$
Conductor $72963$
Order $66$
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(72963, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([44,27,3]))
 
Copy content gp:[g,chi] = znchar(Mod(26674, 72963))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("72963.26674");
 

Basic properties

Modulus: \(72963\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(72963\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(66\)
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 72963.rk

\(\chi_{72963}(43,\cdot)\) \(\chi_{72963}(472,\cdot)\) \(\chi_{72963}(2353,\cdot)\) \(\chi_{72963}(12319,\cdot)\) \(\chi_{72963}(20239,\cdot)\) \(\chi_{72963}(21559,\cdot)\) \(\chi_{72963}(24793,\cdot)\) \(\chi_{72963}(26674,\cdot)\) \(\chi_{72963}(30865,\cdot)\) \(\chi_{72963}(37696,\cdot)\) \(\chi_{72963}(38686,\cdot)\) \(\chi_{72963}(45814,\cdot)\) \(\chi_{72963}(45880,\cdot)\) \(\chi_{72963}(48685,\cdot)\) \(\chi_{72963}(55186,\cdot)\) \(\chi_{72963}(60961,\cdot)\) \(\chi_{72963}(62017,\cdot)\) \(\chi_{72963}(63007,\cdot)\) \(\chi_{72963}(68881,\cdot)\) \(\chi_{72963}(70135,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((32429,36181,4357)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{9}{22}\right),e\left(\frac{1}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 72963 }(26674, a) \) \(1\)\(1\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{21}{22}\right)\)
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)
 
\( \chi_{ 72963 }(26674,a) \;\) at \(\;a = \) e.g. 2