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Properties

Label 380831.59
Modulus $380831$
Conductor $380831$
Order $42680$
Real no
Primitive yes
Minimal yes
Parity odd

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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(380831, base_ring=CyclotomicField(42680)) M = H._module chi = DirichletCharacter(H, M([8536,20855,36740]))
 
Copy content gp:[g,chi] = znchar(Mod(59, 380831))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("380831.59");
 

Basic properties

Modulus: \(380831\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(380831\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(42680\)
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 380831.ip

\(\chi_{380831}(59,\cdot)\) \(\chi_{380831}(86,\cdot)\) \(\chi_{380831}(137,\cdot)\) \(\chi_{380831}(181,\cdot)\) \(\chi_{380831}(202,\cdot)\) \(\chi_{380831}(213,\cdot)\) \(\chi_{380831}(295,\cdot)\) \(\chi_{380831}(313,\cdot)\) \(\chi_{380831}(323,\cdot)\) \(\chi_{380831}(383,\cdot)\) \(\chi_{380831}(476,\cdot)\) \(\chi_{380831}(488,\cdot)\) \(\chi_{380831}(609,\cdot)\) \(\chi_{380831}(636,\cdot)\) \(\chi_{380831}(653,\cdot)\) \(\chi_{380831}(685,\cdot)\) \(\chi_{380831}(697,\cdot)\) \(\chi_{380831}(698,\cdot)\) \(\chi_{380831}(709,\cdot)\) \(\chi_{380831}(742,\cdot)\) \(\chi_{380831}(753,\cdot)\) \(\chi_{380831}(773,\cdot)\) \(\chi_{380831}(830,\cdot)\) \(\chi_{380831}(884,\cdot)\) \(\chi_{380831}(905,\cdot)\) \(\chi_{380831}(918,\cdot)\) \(\chi_{380831}(944,\cdot)\) \(\chi_{380831}(951,\cdot)\) \(\chi_{380831}(1010,\cdot)\) \(\chi_{380831}(1017,\cdot)\) ...

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: $\Q(\zeta_{42680})$
Fixed field: Number field defined by a degree 42680 polynomial (not computed)

Values on generators

\((103864,55628,58741)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{43}{88}\right),e\left(\frac{167}{194}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 380831 }(59, a) \) \(-1\)\(1\)\(e\left(\frac{9379}{10670}\right)\)\(e\left(\frac{15883}{42680}\right)\)\(e\left(\frac{4044}{5335}\right)\)\(e\left(\frac{6697}{21340}\right)\)\(e\left(\frac{10719}{42680}\right)\)\(e\left(\frac{14087}{42680}\right)\)\(e\left(\frac{6797}{10670}\right)\)\(e\left(\frac{15883}{21340}\right)\)\(e\left(\frac{823}{4268}\right)\)\(e\left(\frac{101}{776}\right)\)
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)
 
\( \chi_{ 380831 }(59,a) \;\) at \(\;a = \) e.g. 2