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Properties

Label 3612.2735
Modulus $3612$
Conductor $3612$
Order $42$
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(3612, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([21,21,35,17]))
 
Copy content gp:[g,chi] = znchar(Mod(2735, 3612))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3612.2735");
 

Basic properties

Modulus: \(3612\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3612\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(42\)
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 3612.ga

\(\chi_{3612}(227,\cdot)\) \(\chi_{3612}(803,\cdot)\) \(\chi_{3612}(1151,\cdot)\) \(\chi_{3612}(1739,\cdot)\) \(\chi_{3612}(2327,\cdot)\) \(\chi_{3612}(2399,\cdot)\) \(\chi_{3612}(2411,\cdot)\) \(\chi_{3612}(2567,\cdot)\) \(\chi_{3612}(2651,\cdot)\) \(\chi_{3612}(2735,\cdot)\) \(\chi_{3612}(3071,\cdot)\) \(\chi_{3612}(3503,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.42.163472327560228338143827214127988723755564574245135318725288053985715521353022868433333769809371071888783003311027519488.1

Values on generators

\((1807,1205,3097,3529)\) → \((-1,-1,e\left(\frac{5}{6}\right),e\left(\frac{17}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 3612 }(2735, a) \) \(1\)\(1\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(-1\)
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)
 
\( \chi_{ 3612 }(2735,a) \;\) at \(\;a = \) e.g. 2