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Properties

Label 3415.1682
Modulus $3415$
Conductor $3415$
Order $124$
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(3415, base_ring=CyclotomicField(124)) M = H._module chi = DirichletCharacter(H, M([31,86]))
 
Copy content gp:[g,chi] = znchar(Mod(1682, 3415))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3415.1682");
 

Basic properties

Modulus: \(3415\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3415\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(124\)
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 3415.r

\(\chi_{3415}(37,\cdot)\) \(\chi_{3415}(112,\cdot)\) \(\chi_{3415}(193,\cdot)\) \(\chi_{3415}(292,\cdot)\) \(\chi_{3415}(333,\cdot)\) \(\chi_{3415}(372,\cdot)\) \(\chi_{3415}(433,\cdot)\) \(\chi_{3415}(482,\cdot)\) \(\chi_{3415}(602,\cdot)\) \(\chi_{3415}(607,\cdot)\) \(\chi_{3415}(637,\cdot)\) \(\chi_{3415}(763,\cdot)\) \(\chi_{3415}(807,\cdot)\) \(\chi_{3415}(923,\cdot)\) \(\chi_{3415}(948,\cdot)\) \(\chi_{3415}(952,\cdot)\) \(\chi_{3415}(1008,\cdot)\) \(\chi_{3415}(1113,\cdot)\) \(\chi_{3415}(1123,\cdot)\) \(\chi_{3415}(1138,\cdot)\) \(\chi_{3415}(1228,\cdot)\) \(\chi_{3415}(1262,\cdot)\) \(\chi_{3415}(1357,\cdot)\) \(\chi_{3415}(1363,\cdot)\) \(\chi_{3415}(1403,\cdot)\) \(\chi_{3415}(1477,\cdot)\) \(\chi_{3415}(1478,\cdot)\) \(\chi_{3415}(1658,\cdot)\) \(\chi_{3415}(1682,\cdot)\) \(\chi_{3415}(1702,\cdot)\) ...

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: $\Q(\zeta_{124})$
Fixed field: Number field defined by a degree 124 polynomial (not computed)

Values on generators

\((1367,1371)\) → \((i,e\left(\frac{43}{62}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 3415 }(1682, a) \) \(1\)\(1\)\(-i\)\(e\left(\frac{77}{124}\right)\)\(-1\)\(e\left(\frac{23}{62}\right)\)\(e\left(\frac{5}{124}\right)\)\(i\)\(e\left(\frac{15}{62}\right)\)\(e\left(\frac{41}{62}\right)\)\(e\left(\frac{15}{124}\right)\)\(e\left(\frac{111}{124}\right)\)
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)
 
\( \chi_{ 3415 }(1682,a) \;\) at \(\;a = \) e.g. 2