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Properties

Label 39701.1823
Modulus $39701$
Conductor $39701$
Order $777$
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(39701, base_ring=CyclotomicField(1554)) M = H._module chi = DirichletCharacter(H, M([888,532]))
 
Copy content gp:[g,chi] = znchar(Mod(1823, 39701))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("39701.1823");
 

Basic properties

Modulus: \(39701\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(39701\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(777\)
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 39701.dk

\(\chi_{39701}(248,\cdot)\) \(\chi_{39701}(285,\cdot)\) \(\chi_{39701}(306,\cdot)\) \(\chi_{39701}(343,\cdot)\) \(\chi_{39701}(692,\cdot)\) \(\chi_{39701}(750,\cdot)\) \(\chi_{39701}(803,\cdot)\) \(\chi_{39701}(861,\cdot)\) \(\chi_{39701}(877,\cdot)\) \(\chi_{39701}(935,\cdot)\) \(\chi_{39701}(951,\cdot)\) \(\chi_{39701}(1009,\cdot)\) \(\chi_{39701}(1321,\cdot)\) \(\chi_{39701}(1358,\cdot)\) \(\chi_{39701}(1379,\cdot)\) \(\chi_{39701}(1416,\cdot)\) \(\chi_{39701}(1765,\cdot)\) \(\chi_{39701}(1823,\cdot)\) \(\chi_{39701}(1876,\cdot)\) \(\chi_{39701}(1934,\cdot)\) \(\chi_{39701}(2008,\cdot)\) \(\chi_{39701}(2024,\cdot)\) \(\chi_{39701}(2082,\cdot)\) \(\chi_{39701}(2394,\cdot)\) \(\chi_{39701}(2431,\cdot)\) \(\chi_{39701}(2452,\cdot)\) \(\chi_{39701}(2489,\cdot)\) \(\chi_{39701}(2838,\cdot)\) \(\chi_{39701}(2896,\cdot)\) \(\chi_{39701}(2949,\cdot)\) ...

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: $\Q(\zeta_{777})$
Fixed field: Number field defined by a degree 777 polynomial (not computed)

Values on generators

\((6846,32858)\) → \((e\left(\frac{4}{7}\right),e\left(\frac{38}{111}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 39701 }(1823, a) \) \(1\)\(1\)\(e\left(\frac{710}{777}\right)\)\(e\left(\frac{274}{777}\right)\)\(e\left(\frac{643}{777}\right)\)\(e\left(\frac{115}{777}\right)\)\(e\left(\frac{69}{259}\right)\)\(e\left(\frac{358}{777}\right)\)\(e\left(\frac{192}{259}\right)\)\(e\left(\frac{548}{777}\right)\)\(e\left(\frac{16}{259}\right)\)\(e\left(\frac{102}{259}\right)\)
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)
 
\( \chi_{ 39701 }(1823,a) \;\) at \(\;a = \) e.g. 2