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Properties

Label 25002.1339
Modulus $25002$
Conductor $12501$
Order $1386$
Real no
Primitive no
Minimal yes
Parity odd

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(25002, base_ring=CyclotomicField(1386)) M = H._module chi = DirichletCharacter(H, M([308,159]))
 
Copy content gp:[g,chi] = znchar(Mod(1339, 25002))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("25002.1339");
 

Basic properties

Modulus: \(25002\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(12501\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1386\)
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 \(\chi_{12501}(1339,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 25002.er

\(\chi_{25002}(85,\cdot)\) \(\chi_{25002}(151,\cdot)\) \(\chi_{25002}(175,\cdot)\) \(\chi_{25002}(403,\cdot)\) \(\chi_{25002}(421,\cdot)\) \(\chi_{25002}(511,\cdot)\) \(\chi_{25002}(517,\cdot)\) \(\chi_{25002}(661,\cdot)\) \(\chi_{25002}(715,\cdot)\) \(\chi_{25002}(745,\cdot)\) \(\chi_{25002}(823,\cdot)\) \(\chi_{25002}(853,\cdot)\) \(\chi_{25002}(895,\cdot)\) \(\chi_{25002}(979,\cdot)\) \(\chi_{25002}(1273,\cdot)\) \(\chi_{25002}(1291,\cdot)\) \(\chi_{25002}(1339,\cdot)\) \(\chi_{25002}(1357,\cdot)\) \(\chi_{25002}(1429,\cdot)\) \(\chi_{25002}(1435,\cdot)\)

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: $\Q(\zeta_{693})$
Fixed field: Number field defined by a degree 1386 polynomial (not computed)

Values on generators

\((15743,18523)\) → \((e\left(\frac{2}{9}\right),e\left(\frac{53}{462}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 25002 }(1339, a) \) \(-1\)\(1\)\(e\left(\frac{625}{1386}\right)\)\(e\left(\frac{149}{1386}\right)\)\(e\left(\frac{1139}{1386}\right)\)\(e\left(\frac{11}{126}\right)\)\(e\left(\frac{67}{231}\right)\)\(e\left(\frac{139}{154}\right)\)\(e\left(\frac{1273}{1386}\right)\)\(e\left(\frac{625}{693}\right)\)\(e\left(\frac{430}{693}\right)\)\(e\left(\frac{617}{693}\right)\)
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)
 
\( \chi_{ 25002 }(1339,a) \;\) at \(\;a = \) e.g. 2