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Properties

Label 60270.23647
Modulus $60270$
Conductor $10045$
Order $140$
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(60270, base_ring=CyclotomicField(140)) M = H._module chi = DirichletCharacter(H, M([0,35,60,98]))
 
Copy content gp:[g,chi] = znchar(Mod(23647, 60270))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("60270.23647");
 

Basic properties

Modulus: \(60270\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(10045\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(140\)
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 \(\chi_{10045}(3557,\cdot)\)
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 60270.mz

\(\chi_{60270}(127,\cdot)\) \(\chi_{60270}(2983,\cdot)\) \(\chi_{60270}(4453,\cdot)\) \(\chi_{60270}(6427,\cdot)\) \(\chi_{60270}(7813,\cdot)\) \(\chi_{60270}(7897,\cdot)\) \(\chi_{60270}(8737,\cdot)\) \(\chi_{60270}(11257,\cdot)\) \(\chi_{60270}(11593,\cdot)\) \(\chi_{60270}(13063,\cdot)\) \(\chi_{60270}(13903,\cdot)\) \(\chi_{60270}(15037,\cdot)\) \(\chi_{60270}(16423,\cdot)\) \(\chi_{60270}(16507,\cdot)\) \(\chi_{60270}(19867,\cdot)\) \(\chi_{60270}(20203,\cdot)\) \(\chi_{60270}(21673,\cdot)\) \(\chi_{60270}(22513,\cdot)\) \(\chi_{60270}(23647,\cdot)\) \(\chi_{60270}(25033,\cdot)\) \(\chi_{60270}(25117,\cdot)\) \(\chi_{60270}(25957,\cdot)\) \(\chi_{60270}(28477,\cdot)\) \(\chi_{60270}(31123,\cdot)\) \(\chi_{60270}(32257,\cdot)\) \(\chi_{60270}(33643,\cdot)\) \(\chi_{60270}(33727,\cdot)\) \(\chi_{60270}(34567,\cdot)\) \(\chi_{60270}(37087,\cdot)\) \(\chi_{60270}(37423,\cdot)\) ...

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: $\Q(\zeta_{140})$
Fixed field: Number field defined by a degree 140 polynomial (not computed)

Values on generators

\((40181,48217,24601,8821)\) → \((1,i,e\left(\frac{3}{7}\right),e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(43\)\(47\)
\( \chi_{ 60270 }(23647, a) \) \(-1\)\(1\)\(e\left(\frac{17}{70}\right)\)\(e\left(\frac{83}{140}\right)\)\(e\left(\frac{9}{140}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{33}{140}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{51}{140}\right)\)\(e\left(\frac{73}{140}\right)\)\(e\left(\frac{13}{140}\right)\)
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)
 
\( \chi_{ 60270 }(23647,a) \;\) at \(\;a = \) e.g. 2