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Properties

Label 36260.1349
Modulus $36260$
Conductor $9065$
Order $252$
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(36260, base_ring=CyclotomicField(252)) M = H._module chi = DirichletCharacter(H, M([0,126,102,49]))
 
Copy content gp:[g,chi] = znchar(Mod(1349, 36260))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("36260.1349");
 

Basic properties

Modulus: \(36260\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(9065\)
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 \(\chi_{9065}(1349,\cdot)\)
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 36260.wm

\(\chi_{36260}(89,\cdot)\) \(\chi_{36260}(649,\cdot)\) \(\chi_{36260}(1349,\cdot)\) \(\chi_{36260}(1909,\cdot)\) \(\chi_{36260}(2329,\cdot)\) \(\chi_{36260}(3349,\cdot)\) \(\chi_{36260}(3769,\cdot)\) \(\chi_{36260}(3909,\cdot)\) \(\chi_{36260}(4749,\cdot)\) \(\chi_{36260}(4849,\cdot)\) \(\chi_{36260}(4889,\cdot)\) \(\chi_{36260}(5269,\cdot)\) \(\chi_{36260}(5309,\cdot)\) \(\chi_{36260}(5829,\cdot)\) \(\chi_{36260}(6529,\cdot)\) \(\chi_{36260}(7089,\cdot)\) \(\chi_{36260}(7509,\cdot)\) \(\chi_{36260}(8529,\cdot)\) \(\chi_{36260}(9089,\cdot)\) \(\chi_{36260}(10029,\cdot)\) \(\chi_{36260}(10069,\cdot)\) \(\chi_{36260}(10449,\cdot)\) \(\chi_{36260}(10489,\cdot)\) \(\chi_{36260}(11009,\cdot)\) \(\chi_{36260}(11709,\cdot)\) \(\chi_{36260}(12689,\cdot)\) \(\chi_{36260}(13709,\cdot)\) \(\chi_{36260}(14129,\cdot)\) \(\chi_{36260}(14269,\cdot)\) \(\chi_{36260}(15109,\cdot)\) ...

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

\((18131,21757,14801,34301)\) → \((1,-1,e\left(\frac{17}{42}\right),e\left(\frac{7}{36}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 36260 }(1349, a) \) \(1\)\(1\)\(e\left(\frac{121}{126}\right)\)\(e\left(\frac{58}{63}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{251}{252}\right)\)\(e\left(\frac{247}{252}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{67}{84}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{31}{84}\right)\)\(e\left(\frac{7}{12}\right)\)
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)
 
\( \chi_{ 36260 }(1349,a) \;\) at \(\;a = \) e.g. 2