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Properties

Label 80010.6943
Modulus $80010$
Conductor $40005$
Order $252$
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(80010, base_ring=CyclotomicField(252)) M = H._module chi = DirichletCharacter(H, M([84,189,126,250]))
 
Copy content gp:[g,chi] = znchar(Mod(6943, 80010))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80010.6943");
 

Basic properties

Modulus: \(80010\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(40005\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(252\)
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 \(\chi_{40005}(6943,\cdot)\)
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 80010.bov

\(\chi_{80010}(97,\cdot)\) \(\chi_{80010}(223,\cdot)\) \(\chi_{80010}(727,\cdot)\) \(\chi_{80010}(3037,\cdot)\) \(\chi_{80010}(3793,\cdot)\) \(\chi_{80010}(3877,\cdot)\) \(\chi_{80010}(4297,\cdot)\) \(\chi_{80010}(5137,\cdot)\) \(\chi_{80010}(5263,\cdot)\) \(\chi_{80010}(6187,\cdot)\) \(\chi_{80010}(6523,\cdot)\) \(\chi_{80010}(6817,\cdot)\) \(\chi_{80010}(6943,\cdot)\) \(\chi_{80010}(10723,\cdot)\) \(\chi_{80010}(10807,\cdot)\) \(\chi_{80010}(11437,\cdot)\) \(\chi_{80010}(11563,\cdot)\) \(\chi_{80010}(11983,\cdot)\) \(\chi_{80010}(13957,\cdot)\) \(\chi_{80010}(18787,\cdot)\) \(\chi_{80010}(21937,\cdot)\) \(\chi_{80010}(22273,\cdot)\) \(\chi_{80010}(23197,\cdot)\) \(\chi_{80010}(23827,\cdot)\) \(\chi_{80010}(25423,\cdot)\) \(\chi_{80010}(26347,\cdot)\) \(\chi_{80010}(26557,\cdot)\) \(\chi_{80010}(26977,\cdot)\) \(\chi_{80010}(27943,\cdot)\) \(\chi_{80010}(28237,\cdot)\) ...

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: $\Q(\zeta_{252})$
Fixed field: Number field defined by a degree 252 polynomial (not computed)

Values on generators

\((35561,48007,57151,15751)\) → \((e\left(\frac{1}{3}\right),-i,-1,e\left(\frac{125}{126}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 80010 }(6943, a) \) \(-1\)\(1\)\(e\left(\frac{50}{63}\right)\)\(e\left(\frac{169}{252}\right)\)\(e\left(\frac{239}{252}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{241}{252}\right)\)\(e\left(\frac{59}{63}\right)\)\(e\left(\frac{101}{126}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{67}{126}\right)\)\(e\left(\frac{5}{252}\right)\)
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)
 
\( \chi_{ 80010 }(6943,a) \;\) at \(\;a = \) e.g. 2