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Properties

Label 13870.9811
Modulus $13870$
Conductor $1387$
Order $72$
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(13870, base_ring=CyclotomicField(72)) M = H._module chi = DirichletCharacter(H, M([0,24,35]))
 
Copy content gp:[g,chi] = znchar(Mod(9811, 13870))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("13870.9811");
 

Basic properties

Modulus: \(13870\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1387\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(72\)
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 \(\chi_{1387}(102,\cdot)\)
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 13870.oy

\(\chi_{13870}(11,\cdot)\) \(\chi_{13870}(391,\cdot)\) \(\chi_{13870}(691,\cdot)\) \(\chi_{13870}(1261,\cdot)\) \(\chi_{13870}(2291,\cdot)\) \(\chi_{13870}(3051,\cdot)\) \(\chi_{13870}(3811,\cdot)\) \(\chi_{13870}(4571,\cdot)\) \(\chi_{13870}(4871,\cdot)\) \(\chi_{13870}(5441,\cdot)\) \(\chi_{13870}(6471,\cdot)\) \(\chi_{13870}(6851,\cdot)\) \(\chi_{13870}(7041,\cdot)\) \(\chi_{13870}(8181,\cdot)\) \(\chi_{13870}(8291,\cdot)\) \(\chi_{13870}(8481,\cdot)\) \(\chi_{13870}(9431,\cdot)\) \(\chi_{13870}(9811,\cdot)\) \(\chi_{13870}(10191,\cdot)\) \(\chi_{13870}(10571,\cdot)\) \(\chi_{13870}(11521,\cdot)\) \(\chi_{13870}(11711,\cdot)\) \(\chi_{13870}(12551,\cdot)\) \(\chi_{13870}(13691,\cdot)\)

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: $\Q(\zeta_{72})$
Fixed field: Number field defined by a degree 72 polynomial

Values on generators

\((11097,8761,3801)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{35}{72}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 13870 }(9811, a) \) \(-1\)\(1\)\(i\)\(e\left(\frac{1}{24}\right)\)\(-1\)\(e\left(\frac{53}{72}\right)\)\(e\left(\frac{25}{72}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{1}{36}\right)\)\(-i\)\(e\left(\frac{49}{72}\right)\)
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)
 
\( \chi_{ 13870 }(9811,a) \;\) at \(\;a = \) e.g. 2