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Properties

Label 3616.1155
Modulus $3616$
Conductor $3616$
Order $56$
Real no
Primitive yes
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(3616, base_ring=CyclotomicField(56)) M = H._module chi = DirichletCharacter(H, M([28,21,27]))
 
Copy content gp:[g,chi] = znchar(Mod(1155, 3616))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3616.1155");
 

Basic properties

Modulus: \(3616\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3616\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(56\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 3616.dq

\(\chi_{3616}(11,\cdot)\) \(\chi_{3616}(627,\cdot)\) \(\chi_{3616}(667,\cdot)\) \(\chi_{3616}(739,\cdot)\) \(\chi_{3616}(1099,\cdot)\) \(\chi_{3616}(1155,\cdot)\) \(\chi_{3616}(1171,\cdot)\) \(\chi_{3616}(1315,\cdot)\) \(\chi_{3616}(1331,\cdot)\) \(\chi_{3616}(1419,\cdot)\) \(\chi_{3616}(1659,\cdot)\) \(\chi_{3616}(1747,\cdot)\) \(\chi_{3616}(1859,\cdot)\) \(\chi_{3616}(1947,\cdot)\) \(\chi_{3616}(2043,\cdot)\) \(\chi_{3616}(2251,\cdot)\) \(\chi_{3616}(2347,\cdot)\) \(\chi_{3616}(2499,\cdot)\) \(\chi_{3616}(2635,\cdot)\) \(\chi_{3616}(2803,\cdot)\) \(\chi_{3616}(2875,\cdot)\) \(\chi_{3616}(3195,\cdot)\) \(\chi_{3616}(3299,\cdot)\) \(\chi_{3616}(3603,\cdot)\)

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: $\Q(\zeta_{56})$
Fixed field: Number field defined by a degree 56 polynomial

Values on generators

\((3391,453,3393)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{27}{56}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 3616 }(1155, a) \) \(-1\)\(1\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{3}{56}\right)\)\(e\left(\frac{13}{56}\right)\)\(-1\)\(e\left(\frac{51}{56}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{3}{14}\right)\)
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)
 
\( \chi_{ 3616 }(1155,a) \;\) at \(\;a = \) e.g. 2