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Properties

Label 644809.346
Modulus $644809$
Conductor $644809$
Order $144540$
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(644809, base_ring=CyclotomicField(144540)) M = H._module chi = DirichletCharacter(H, M([136656,75955]))
 
Copy content gp:[g,chi] = znchar(Mod(346, 644809))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("644809.346");
 

Basic properties

Modulus: \(644809\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(644809\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(144540\)
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 644809.hh

\(\chi_{644809}(25,\cdot)\) \(\chi_{644809}(38,\cdot)\) \(\chi_{644809}(48,\cdot)\) \(\chi_{644809}(92,\cdot)\) \(\chi_{644809}(108,\cdot)\) \(\chi_{644809}(152,\cdot)\) \(\chi_{644809}(158,\cdot)\) \(\chi_{644809}(169,\cdot)\) \(\chi_{644809}(181,\cdot)\) \(\chi_{644809}(196,\cdot)\) \(\chi_{644809}(207,\cdot)\) \(\chi_{644809}(213,\cdot)\) \(\chi_{644809}(225,\cdot)\) \(\chi_{644809}(257,\cdot)\) \(\chi_{644809}(267,\cdot)\) \(\chi_{644809}(273,\cdot)\) \(\chi_{644809}(280,\cdot)\) \(\chi_{644809}(311,\cdot)\) \(\chi_{644809}(317,\cdot)\) \(\chi_{644809}(346,\cdot)\) \(\chi_{644809}(377,\cdot)\) \(\chi_{644809}(388,\cdot)\) \(\chi_{644809}(400,\cdot)\) \(\chi_{644809}(432,\cdot)\) \(\chi_{644809}(476,\cdot)\) \(\chi_{644809}(488,\cdot)\) \(\chi_{644809}(499,\cdot)\) \(\chi_{644809}(559,\cdot)\) \(\chi_{644809}(603,\cdot)\) \(\chi_{644809}(609,\cdot)\) ...

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: $\Q(\zeta_{144540})$
Fixed field: Number field defined by a degree 144540 polynomial (not computed)

Values on generators

\((516914,511589)\) → \((e\left(\frac{52}{55}\right),e\left(\frac{1381}{2628}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 644809 }(346, a) \) \(1\)\(1\)\(e\left(\frac{7379}{36135}\right)\)\(e\left(\frac{1883}{2190}\right)\)\(e\left(\frac{14758}{36135}\right)\)\(e\left(\frac{70699}{144540}\right)\)\(e\left(\frac{4627}{72270}\right)\)\(e\left(\frac{9269}{48180}\right)\)\(e\left(\frac{7379}{12045}\right)\)\(e\left(\frac{788}{1095}\right)\)\(e\left(\frac{2227}{3212}\right)\)\(e\left(\frac{3877}{14454}\right)\)
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)
 
\( \chi_{ 644809 }(346,a) \;\) at \(\;a = \) e.g. 2