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Properties

Label 35392.173
Modulus $35392$
Conductor $35392$
Order $624$
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(35392, base_ring=CyclotomicField(624)) M = H._module chi = DirichletCharacter(H, M([0,273,520,504]))
 
Copy content gp:[g,chi] = znchar(Mod(173, 35392))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("35392.173");
 

Basic properties

Modulus: \(35392\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(35392\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(624\)
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 35392.ox

\(\chi_{35392}(61,\cdot)\) \(\chi_{35392}(173,\cdot)\) \(\chi_{35392}(229,\cdot)\) \(\chi_{35392}(453,\cdot)\) \(\chi_{35392}(565,\cdot)\) \(\chi_{35392}(1333,\cdot)\) \(\chi_{35392}(1613,\cdot)\) \(\chi_{35392}(1965,\cdot)\) \(\chi_{35392}(2229,\cdot)\) \(\chi_{35392}(2245,\cdot)\) \(\chi_{35392}(2397,\cdot)\) \(\chi_{35392}(2621,\cdot)\) \(\chi_{35392}(2861,\cdot)\) \(\chi_{35392}(2901,\cdot)\) \(\chi_{35392}(3029,\cdot)\) \(\chi_{35392}(3253,\cdot)\) \(\chi_{35392}(3517,\cdot)\) \(\chi_{35392}(3533,\cdot)\) \(\chi_{35392}(3853,\cdot)\) \(\chi_{35392}(3965,\cdot)\) \(\chi_{35392}(4021,\cdot)\) \(\chi_{35392}(4149,\cdot)\) \(\chi_{35392}(4245,\cdot)\) \(\chi_{35392}(4357,\cdot)\) \(\chi_{35392}(4485,\cdot)\) \(\chi_{35392}(4597,\cdot)\) \(\chi_{35392}(4653,\cdot)\) \(\chi_{35392}(4877,\cdot)\) \(\chi_{35392}(4989,\cdot)\) \(\chi_{35392}(5757,\cdot)\) ...

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: $\Q(\zeta_{624})$
Fixed field: Number field defined by a degree 624 polynomial (not computed)

Values on generators

\((7743,19909,5057,2689)\) → \((1,e\left(\frac{7}{16}\right),e\left(\frac{5}{6}\right),e\left(\frac{21}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 35392 }(173, a) \) \(1\)\(1\)\(e\left(\frac{595}{624}\right)\)\(e\left(\frac{425}{624}\right)\)\(e\left(\frac{283}{312}\right)\)\(e\left(\frac{277}{624}\right)\)\(e\left(\frac{109}{208}\right)\)\(e\left(\frac{33}{52}\right)\)\(e\left(\frac{7}{156}\right)\)\(e\left(\frac{47}{624}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{113}{312}\right)\)
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)
 
\( \chi_{ 35392 }(173,a) \;\) at \(\;a = \) e.g. 2