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Properties

Label 66924.349
Modulus $66924$
Conductor $16731$
Order $780$
Real no
Primitive no
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(66924, base_ring=CyclotomicField(780)) M = H._module chi = DirichletCharacter(H, M([0,520,234,515]))
 
Copy content gp:[g,chi] = znchar(Mod(349, 66924))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("66924.349");
 

Basic properties

Modulus: \(66924\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(16731\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(780\)
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: no, induced from \(\chi_{16731}(349,\cdot)\)
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 66924.rt

\(\chi_{66924}(85,\cdot)\) \(\chi_{66924}(349,\cdot)\) \(\chi_{66924}(409,\cdot)\) \(\chi_{66924}(457,\cdot)\) \(\chi_{66924}(877,\cdot)\) \(\chi_{66924}(1393,\cdot)\) \(\chi_{66924}(1861,\cdot)\) \(\chi_{66924}(2329,\cdot)\) \(\chi_{66924}(3625,\cdot)\) \(\chi_{66924}(4153,\cdot)\) \(\chi_{66924}(4297,\cdot)\) \(\chi_{66924}(4561,\cdot)\) \(\chi_{66924}(4765,\cdot)\) \(\chi_{66924}(5029,\cdot)\) \(\chi_{66924}(5233,\cdot)\) \(\chi_{66924}(5557,\cdot)\) \(\chi_{66924}(5605,\cdot)\) \(\chi_{66924}(6025,\cdot)\) \(\chi_{66924}(6541,\cdot)\) \(\chi_{66924}(7477,\cdot)\) \(\chi_{66924}(8509,\cdot)\) \(\chi_{66924}(8773,\cdot)\) \(\chi_{66924}(9301,\cdot)\) \(\chi_{66924}(9709,\cdot)\) \(\chi_{66924}(9913,\cdot)\) \(\chi_{66924}(10177,\cdot)\) \(\chi_{66924}(10237,\cdot)\) \(\chi_{66924}(10381,\cdot)\) \(\chi_{66924}(10645,\cdot)\) \(\chi_{66924}(10705,\cdot)\) ...

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: $\Q(\zeta_{780})$
Fixed field: Number field defined by a degree 780 polynomial (not computed)

Values on generators

\((33463,37181,6085,40393)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{3}{10}\right),e\left(\frac{103}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)\(37\)
\( \chi_{ 66924 }(349, a) \) \(1\)\(1\)\(e\left(\frac{371}{780}\right)\)\(e\left(\frac{323}{780}\right)\)\(e\left(\frac{19}{195}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{371}{390}\right)\)\(e\left(\frac{23}{130}\right)\)\(e\left(\frac{779}{780}\right)\)\(e\left(\frac{347}{390}\right)\)\(e\left(\frac{233}{780}\right)\)
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)
 
\( \chi_{ 66924 }(349,a) \;\) at \(\;a = \) e.g. 2