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Properties

Label 717409.13
Modulus $717409$
Conductor $717409$
Order $93170$
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(717409, base_ring=CyclotomicField(93170)) M = H._module chi = DirichletCharacter(H, M([73205,8407]))
 
Copy content gp:[g,chi] = znchar(Mod(13, 717409))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("717409.13");
 

Basic properties

Modulus: \(717409\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(717409\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(93170\)
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 717409.et

\(\chi_{717409}(6,\cdot)\) \(\chi_{717409}(13,\cdot)\) \(\chi_{717409}(41,\cdot)\) \(\chi_{717409}(62,\cdot)\) \(\chi_{717409}(83,\cdot)\) \(\chi_{717409}(90,\cdot)\) \(\chi_{717409}(139,\cdot)\) \(\chi_{717409}(160,\cdot)\) \(\chi_{717409}(167,\cdot)\) \(\chi_{717409}(216,\cdot)\) \(\chi_{717409}(237,\cdot)\) \(\chi_{717409}(272,\cdot)\) \(\chi_{717409}(314,\cdot)\) \(\chi_{717409}(321,\cdot)\) \(\chi_{717409}(349,\cdot)\) \(\chi_{717409}(370,\cdot)\) \(\chi_{717409}(398,\cdot)\) \(\chi_{717409}(426,\cdot)\) \(\chi_{717409}(447,\cdot)\) \(\chi_{717409}(468,\cdot)\) \(\chi_{717409}(503,\cdot)\) \(\chi_{717409}(545,\cdot)\) \(\chi_{717409}(552,\cdot)\) \(\chi_{717409}(580,\cdot)\) \(\chi_{717409}(601,\cdot)\) \(\chi_{717409}(622,\cdot)\) \(\chi_{717409}(629,\cdot)\) \(\chi_{717409}(657,\cdot)\) \(\chi_{717409}(678,\cdot)\) \(\chi_{717409}(706,\cdot)\) ...

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: $\Q(\zeta_{46585})$
Fixed field: Number field defined by a degree 93170 polynomial (not computed)

Values on generators

\((571000,73207)\) → \((e\left(\frac{11}{14}\right),e\left(\frac{1201}{13310}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 717409 }(13, a) \) \(1\)\(1\)\(e\left(\frac{48337}{93170}\right)\)\(e\left(\frac{6851}{8470}\right)\)\(e\left(\frac{1752}{46585}\right)\)\(e\left(\frac{87023}{93170}\right)\)\(e\left(\frac{15264}{46585}\right)\)\(e\left(\frac{51841}{93170}\right)\)\(e\left(\frac{2616}{4235}\right)\)\(e\left(\frac{4219}{9317}\right)\)\(e\left(\frac{15773}{18634}\right)\)\(e\left(\frac{13896}{46585}\right)\)
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)
 
\( \chi_{ 717409 }(13,a) \;\) at \(\;a = \) e.g. 2