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Properties

Label 84042.4775
Modulus $84042$
Conductor $6003$
Order $924$
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(84042, base_ring=CyclotomicField(924)) M = H._module chi = DirichletCharacter(H, M([770,0,882,297]))
 
Copy content gp:[g,chi] = znchar(Mod(4775, 84042))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("84042.4775");
 

Basic properties

Modulus: \(84042\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6003\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(924\)
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 \(\chi_{6003}(4775,\cdot)\)
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 84042.se

\(\chi_{84042}(113,\cdot)\) \(\chi_{84042}(155,\cdot)\) \(\chi_{84042}(617,\cdot)\) \(\chi_{84042}(659,\cdot)\) \(\chi_{84042}(743,\cdot)\) \(\chi_{84042}(1121,\cdot)\) \(\chi_{84042}(1667,\cdot)\) \(\chi_{84042}(1877,\cdot)\) \(\chi_{84042}(2045,\cdot)\) \(\chi_{84042}(2549,\cdot)\) \(\chi_{84042}(2885,\cdot)\) \(\chi_{84042}(3053,\cdot)\) \(\chi_{84042}(3179,\cdot)\) \(\chi_{84042}(3263,\cdot)\) \(\chi_{84042}(3557,\cdot)\) \(\chi_{84042}(3809,\cdot)\) \(\chi_{84042}(4145,\cdot)\) \(\chi_{84042}(4775,\cdot)\) \(\chi_{84042}(4817,\cdot)\) \(\chi_{84042}(5531,\cdot)\) \(\chi_{84042}(6539,\cdot)\) \(\chi_{84042}(6707,\cdot)\) \(\chi_{84042}(6917,\cdot)\) \(\chi_{84042}(7421,\cdot)\) \(\chi_{84042}(7463,\cdot)\) \(\chi_{84042}(7841,\cdot)\) \(\chi_{84042}(8093,\cdot)\) \(\chi_{84042}(8429,\cdot)\) \(\chi_{84042}(8471,\cdot)\) \(\chi_{84042}(8681,\cdot)\) ...

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: $\Q(\zeta_{924})$
Fixed field: Number field defined by a degree 924 polynomial (not computed)

Values on generators

\((18677,24013,51157,40573)\) → \((e\left(\frac{5}{6}\right),1,e\left(\frac{21}{22}\right),e\left(\frac{9}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(25\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 84042 }(4775, a) \) \(-1\)\(1\)\(e\left(\frac{89}{462}\right)\)\(e\left(\frac{425}{924}\right)\)\(e\left(\frac{377}{462}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{65}{308}\right)\)\(e\left(\frac{89}{231}\right)\)\(e\left(\frac{661}{924}\right)\)\(e\left(\frac{3}{308}\right)\)\(e\left(\frac{115}{132}\right)\)\(e\left(\frac{263}{924}\right)\)
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)
 
\( \chi_{ 84042 }(4775,a) \;\) at \(\;a = \) e.g. 2