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Properties

Label 28042.1961
Modulus $28042$
Conductor $2003$
Order $286$
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(28042, base_ring=CyclotomicField(286)) M = H._module chi = DirichletCharacter(H, M([0,115]))
 
Copy content gp:[g,chi] = znchar(Mod(1961, 28042))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("28042.1961");
 

Basic properties

Modulus: \(28042\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2003\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(286\)
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 \(\chi_{2003}(1961,\cdot)\)
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 28042.bt

\(\chi_{28042}(71,\cdot)\) \(\chi_{28042}(239,\cdot)\) \(\chi_{28042}(813,\cdot)\) \(\chi_{28042}(1079,\cdot)\) \(\chi_{28042}(1275,\cdot)\) \(\chi_{28042}(1331,\cdot)\) \(\chi_{28042}(1821,\cdot)\) \(\chi_{28042}(1835,\cdot)\) \(\chi_{28042}(1961,\cdot)\) \(\chi_{28042}(2087,\cdot)\) \(\chi_{28042}(2339,\cdot)\) \(\chi_{28042}(2367,\cdot)\) \(\chi_{28042}(2465,\cdot)\) \(\chi_{28042}(2969,\cdot)\) \(\chi_{28042}(3347,\cdot)\) \(\chi_{28042}(3459,\cdot)\) \(\chi_{28042}(3529,\cdot)\) \(\chi_{28042}(3851,\cdot)\) \(\chi_{28042}(3865,\cdot)\) \(\chi_{28042}(3893,\cdot)\) \(\chi_{28042}(4201,\cdot)\) \(\chi_{28042}(4383,\cdot)\) \(\chi_{28042}(4817,\cdot)\) \(\chi_{28042}(4985,\cdot)\) \(\chi_{28042}(5531,\cdot)\) \(\chi_{28042}(5825,\cdot)\) \(\chi_{28042}(5867,\cdot)\) \(\chi_{28042}(5993,\cdot)\) \(\chi_{28042}(6357,\cdot)\) \(\chi_{28042}(6721,\cdot)\) ...

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: $\Q(\zeta_{143})$
Fixed field: Number field defined by a degree 286 polynomial (not computed)

Values on generators

\((4007,20035)\) → \((1,e\left(\frac{115}{286}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 28042 }(1961, a) \) \(-1\)\(1\)\(e\left(\frac{18}{143}\right)\)\(e\left(\frac{115}{286}\right)\)\(e\left(\frac{36}{143}\right)\)\(e\left(\frac{201}{286}\right)\)\(e\left(\frac{105}{143}\right)\)\(e\left(\frac{151}{286}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{47}{143}\right)\)\(e\left(\frac{19}{286}\right)\)\(e\left(\frac{115}{143}\right)\)
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)
 
\( \chi_{ 28042 }(1961,a) \;\) at \(\;a = \) e.g. 2