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Properties

Label 1113.1102
Modulus $1113$
Conductor $371$
Order $78$
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(1113, base_ring=CyclotomicField(78)) M = H._module chi = DirichletCharacter(H, M([0,13,48]))
 
Copy content gp:[g,chi] = znchar(Mod(1102, 1113))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1113.1102");
 

Basic properties

Modulus: \(1113\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(371\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(78\)
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 \(\chi_{371}(360,\cdot)\)
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 1113.bq

\(\chi_{1113}(10,\cdot)\) \(\chi_{1113}(187,\cdot)\) \(\chi_{1113}(208,\cdot)\) \(\chi_{1113}(334,\cdot)\) \(\chi_{1113}(346,\cdot)\) \(\chi_{1113}(367,\cdot)\) \(\chi_{1113}(418,\cdot)\) \(\chi_{1113}(439,\cdot)\) \(\chi_{1113}(460,\cdot)\) \(\chi_{1113}(493,\cdot)\) \(\chi_{1113}(523,\cdot)\) \(\chi_{1113}(577,\cdot)\) \(\chi_{1113}(598,\cdot)\) \(\chi_{1113}(607,\cdot)\) \(\chi_{1113}(619,\cdot)\) \(\chi_{1113}(649,\cdot)\) \(\chi_{1113}(682,\cdot)\) \(\chi_{1113}(733,\cdot)\) \(\chi_{1113}(766,\cdot)\) \(\chi_{1113}(808,\cdot)\) \(\chi_{1113}(892,\cdot)\) \(\chi_{1113}(943,\cdot)\) \(\chi_{1113}(964,\cdot)\) \(\chi_{1113}(1102,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((743,955,1009)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{8}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 1113 }(1102, a) \) \(-1\)\(1\)\(e\left(\frac{37}{39}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{59}{78}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{55}{78}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{7}{26}\right)\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{25}{78}\right)\)\(e\left(\frac{47}{78}\right)\)
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)
 
\( \chi_{ 1113 }(1102,a) \;\) at \(\;a = \) e.g. 2