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Properties

Label 2622.1087
Modulus $2622$
Conductor $437$
Order $99$
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(2622, base_ring=CyclotomicField(198)) M = H._module chi = DirichletCharacter(H, M([0,22,162]))
 
Copy content gp:[g,chi] = znchar(Mod(1087, 2622))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2622.1087");
 

Basic properties

Modulus: \(2622\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(437\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(99\)
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 \(\chi_{437}(213,\cdot)\)
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 2622.bo

\(\chi_{2622}(25,\cdot)\) \(\chi_{2622}(55,\cdot)\) \(\chi_{2622}(73,\cdot)\) \(\chi_{2622}(85,\cdot)\) \(\chi_{2622}(169,\cdot)\) \(\chi_{2622}(187,\cdot)\) \(\chi_{2622}(271,\cdot)\) \(\chi_{2622}(289,\cdot)\) \(\chi_{2622}(301,\cdot)\) \(\chi_{2622}(397,\cdot)\) \(\chi_{2622}(403,\cdot)\) \(\chi_{2622}(427,\cdot)\) \(\chi_{2622}(499,\cdot)\) \(\chi_{2622}(541,\cdot)\) \(\chi_{2622}(625,\cdot)\) \(\chi_{2622}(739,\cdot)\) \(\chi_{2622}(745,\cdot)\) \(\chi_{2622}(823,\cdot)\) \(\chi_{2622}(841,\cdot)\) \(\chi_{2622}(853,\cdot)\) \(\chi_{2622}(859,\cdot)\) \(\chi_{2622}(883,\cdot)\) \(\chi_{2622}(955,\cdot)\) \(\chi_{2622}(997,\cdot)\) \(\chi_{2622}(1051,\cdot)\) \(\chi_{2622}(1087,\cdot)\) \(\chi_{2622}(1099,\cdot)\) \(\chi_{2622}(1225,\cdot)\) \(\chi_{2622}(1297,\cdot)\) \(\chi_{2622}(1315,\cdot)\) ...

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: $\Q(\zeta_{99})$
Fixed field: Number field defined by a degree 99 polynomial

Values on generators

\((875,553,2167)\) → \((1,e\left(\frac{1}{9}\right),e\left(\frac{9}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(25\)\(29\)\(31\)\(35\)\(37\)
\( \chi_{ 2622 }(1087, a) \) \(1\)\(1\)\(e\left(\frac{59}{99}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{1}{99}\right)\)\(e\left(\frac{83}{99}\right)\)\(e\left(\frac{19}{99}\right)\)\(e\left(\frac{61}{99}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{80}{99}\right)\)\(e\left(\frac{2}{11}\right)\)
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)
 
\( \chi_{ 2622 }(1087,a) \;\) at \(\;a = \) e.g. 2