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Properties

Label 1617.743
Modulus $1617$
Conductor $1617$
Order $70$
Real no
Primitive yes
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(1617, base_ring=CyclotomicField(70)) M = H._module chi = DirichletCharacter(H, M([35,60,63]))
 
Copy content gp:[g,chi] = znchar(Mod(743, 1617))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1617.743");
 

Basic properties

Modulus: \(1617\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1617\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(70\)
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: yes
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 1617.cc

\(\chi_{1617}(8,\cdot)\) \(\chi_{1617}(29,\cdot)\) \(\chi_{1617}(134,\cdot)\) \(\chi_{1617}(239,\cdot)\) \(\chi_{1617}(260,\cdot)\) \(\chi_{1617}(281,\cdot)\) \(\chi_{1617}(365,\cdot)\) \(\chi_{1617}(470,\cdot)\) \(\chi_{1617}(512,\cdot)\) \(\chi_{1617}(596,\cdot)\) \(\chi_{1617}(701,\cdot)\) \(\chi_{1617}(722,\cdot)\) \(\chi_{1617}(743,\cdot)\) \(\chi_{1617}(827,\cdot)\) \(\chi_{1617}(953,\cdot)\) \(\chi_{1617}(974,\cdot)\) \(\chi_{1617}(1058,\cdot)\) \(\chi_{1617}(1163,\cdot)\) \(\chi_{1617}(1184,\cdot)\) \(\chi_{1617}(1205,\cdot)\) \(\chi_{1617}(1289,\cdot)\) \(\chi_{1617}(1394,\cdot)\) \(\chi_{1617}(1415,\cdot)\) \(\chi_{1617}(1436,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((1079,199,442)\) → \((-1,e\left(\frac{6}{7}\right),e\left(\frac{9}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 1617 }(743, a) \) \(1\)\(1\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{67}{70}\right)\)\(e\left(\frac{2}{35}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{13}{70}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{1}{35}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{23}{70}\right)\)
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)
 
\( \chi_{ 1617 }(743,a) \;\) at \(\;a = \) e.g. 2