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Properties

Label 2115.1076
Modulus $2115$
Conductor $423$
Order $138$
Real no
Primitive no
Minimal yes
Parity odd

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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(2115, base_ring=CyclotomicField(138)) M = H._module chi = DirichletCharacter(H, M([115,0,72]))
 
Copy content gp:[g,chi] = znchar(Mod(1076, 2115))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2115.1076");
 

Basic properties

Modulus: \(2115\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(423\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(138\)
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 \(\chi_{423}(230,\cdot)\)
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 2115.bn

\(\chi_{2115}(56,\cdot)\) \(\chi_{2115}(101,\cdot)\) \(\chi_{2115}(131,\cdot)\) \(\chi_{2115}(191,\cdot)\) \(\chi_{2115}(356,\cdot)\) \(\chi_{2115}(371,\cdot)\) \(\chi_{2115}(401,\cdot)\) \(\chi_{2115}(491,\cdot)\) \(\chi_{2115}(506,\cdot)\) \(\chi_{2115}(551,\cdot)\) \(\chi_{2115}(581,\cdot)\) \(\chi_{2115}(596,\cdot)\) \(\chi_{2115}(686,\cdot)\) \(\chi_{2115}(761,\cdot)\) \(\chi_{2115}(776,\cdot)\) \(\chi_{2115}(806,\cdot)\) \(\chi_{2115}(896,\cdot)\) \(\chi_{2115}(911,\cdot)\) \(\chi_{2115}(956,\cdot)\) \(\chi_{2115}(1001,\cdot)\) \(\chi_{2115}(1046,\cdot)\) \(\chi_{2115}(1076,\cdot)\) \(\chi_{2115}(1136,\cdot)\) \(\chi_{2115}(1181,\cdot)\) \(\chi_{2115}(1211,\cdot)\) \(\chi_{2115}(1226,\cdot)\) \(\chi_{2115}(1256,\cdot)\) \(\chi_{2115}(1271,\cdot)\) \(\chi_{2115}(1301,\cdot)\) \(\chi_{2115}(1391,\cdot)\) ...

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: $\Q(\zeta_{69})$
Fixed field: Number field defined by a degree 138 polynomial (not computed)

Values on generators

\((236,847,2026)\) → \((e\left(\frac{5}{6}\right),1,e\left(\frac{12}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 2115 }(1076, a) \) \(-1\)\(1\)\(e\left(\frac{31}{138}\right)\)\(e\left(\frac{31}{69}\right)\)\(e\left(\frac{2}{69}\right)\)\(e\left(\frac{31}{46}\right)\)\(e\left(\frac{67}{138}\right)\)\(e\left(\frac{28}{69}\right)\)\(e\left(\frac{35}{138}\right)\)\(e\left(\frac{62}{69}\right)\)\(e\left(\frac{39}{46}\right)\)\(e\left(\frac{11}{23}\right)\)
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)
 
\( \chi_{ 2115 }(1076,a) \;\) at \(\;a = \) e.g. 2