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Properties

Label 99450.35161
Modulus $99450$
Conductor $49725$
Order $240$
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(99450, base_ring=CyclotomicField(240)) M = H._module chi = DirichletCharacter(H, M([160,192,160,75]))
 
Copy content gp:[g,chi] = znchar(Mod(35161, 99450))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("99450.35161");
 

Basic properties

Modulus: \(99450\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(49725\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(240\)
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 \(\chi_{49725}(35161,\cdot)\)
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 99450.byz

\(\chi_{99450}(61,\cdot)\) \(\chi_{99450}(211,\cdot)\) \(\chi_{99450}(1231,\cdot)\) \(\chi_{99450}(4891,\cdot)\) \(\chi_{99450}(5911,\cdot)\) \(\chi_{99450}(7231,\cdot)\) \(\chi_{99450}(9421,\cdot)\) \(\chi_{99450}(10741,\cdot)\) \(\chi_{99450}(11761,\cdot)\) \(\chi_{99450}(11911,\cdot)\) \(\chi_{99450}(12931,\cdot)\) \(\chi_{99450}(15271,\cdot)\) \(\chi_{99450}(18931,\cdot)\) \(\chi_{99450}(21121,\cdot)\) \(\chi_{99450}(24781,\cdot)\) \(\chi_{99450}(27121,\cdot)\) \(\chi_{99450}(28141,\cdot)\) \(\chi_{99450}(28291,\cdot)\) \(\chi_{99450}(29311,\cdot)\) \(\chi_{99450}(30631,\cdot)\) \(\chi_{99450}(32821,\cdot)\) \(\chi_{99450}(34141,\cdot)\) \(\chi_{99450}(35161,\cdot)\) \(\chi_{99450}(38821,\cdot)\) \(\chi_{99450}(39841,\cdot)\) \(\chi_{99450}(39991,\cdot)\) \(\chi_{99450}(41011,\cdot)\) \(\chi_{99450}(44671,\cdot)\) \(\chi_{99450}(45691,\cdot)\) \(\chi_{99450}(47011,\cdot)\) ...

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: $\Q(\zeta_{240})$
Fixed field: Number field defined by a degree 240 polynomial (not computed)

Values on generators

\((44201,67627,84151,5851)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{4}{5}\right),e\left(\frac{2}{3}\right),e\left(\frac{5}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 99450 }(35161, a) \) \(-1\)\(1\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{77}{240}\right)\)\(e\left(\frac{13}{120}\right)\)\(e\left(\frac{39}{80}\right)\)\(e\left(\frac{239}{240}\right)\)\(e\left(\frac{131}{240}\right)\)\(e\left(\frac{43}{240}\right)\)\(e\left(\frac{51}{80}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{31}{60}\right)\)
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)
 
\( \chi_{ 99450 }(35161,a) \;\) at \(\;a = \) e.g. 2