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Properties

Label 3636.1603
Modulus $3636$
Conductor $404$
Order $50$
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(3636, base_ring=CyclotomicField(50)) M = H._module chi = DirichletCharacter(H, M([25,0,8]))
 
Copy content gp:[g,chi] = znchar(Mod(1603, 3636))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3636.1603");
 

Basic properties

Modulus: \(3636\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(404\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(50\)
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 \(\chi_{404}(391,\cdot)\)
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 3636.bt

\(\chi_{3636}(19,\cdot)\) \(\chi_{3636}(559,\cdot)\) \(\chi_{3636}(631,\cdot)\) \(\chi_{3636}(703,\cdot)\) \(\chi_{3636}(775,\cdot)\) \(\chi_{3636}(1135,\cdot)\) \(\chi_{3636}(1243,\cdot)\) \(\chi_{3636}(1495,\cdot)\) \(\chi_{3636}(1531,\cdot)\) \(\chi_{3636}(1567,\cdot)\) \(\chi_{3636}(1603,\cdot)\) \(\chi_{3636}(1855,\cdot)\) \(\chi_{3636}(1999,\cdot)\) \(\chi_{3636}(2179,\cdot)\) \(\chi_{3636}(2503,\cdot)\) \(\chi_{3636}(2899,\cdot)\) \(\chi_{3636}(3007,\cdot)\) \(\chi_{3636}(3187,\cdot)\) \(\chi_{3636}(3223,\cdot)\) \(\chi_{3636}(3439,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Values on generators

\((1819,3233,3133)\) → \((-1,1,e\left(\frac{4}{25}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 3636 }(1603, a) \) \(-1\)\(1\)\(e\left(\frac{21}{25}\right)\)\(e\left(\frac{47}{50}\right)\)\(e\left(\frac{29}{50}\right)\)\(e\left(\frac{14}{25}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{43}{50}\right)\)\(e\left(\frac{13}{50}\right)\)\(e\left(\frac{17}{25}\right)\)\(e\left(\frac{14}{25}\right)\)\(e\left(\frac{47}{50}\right)\)
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)
 
\( \chi_{ 3636 }(1603,a) \;\) at \(\;a = \) e.g. 2