↑

Properties

Label 3010.2507
Modulus $3010$
Conductor $215$
Order $84$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3010, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([21,0,64]))
 
Copy content gp:[g,chi] = znchar(Mod(2507, 3010))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3010.2507");
 

Basic properties

Modulus: \(3010\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(215\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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 \(\chi_{215}(142,\cdot)\)
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 3010.ea

\(\chi_{3010}(57,\cdot)\) \(\chi_{3010}(197,\cdot)\) \(\chi_{3010}(253,\cdot)\) \(\chi_{3010}(267,\cdot)\) \(\chi_{3010}(533,\cdot)\) \(\chi_{3010}(547,\cdot)\) \(\chi_{3010}(617,\cdot)\) \(\chi_{3010}(827,\cdot)\) \(\chi_{3010}(883,\cdot)\) \(\chi_{3010}(1303,\cdot)\) \(\chi_{3010}(1373,\cdot)\) \(\chi_{3010}(1443,\cdot)\) \(\chi_{3010}(1457,\cdot)\) \(\chi_{3010}(1737,\cdot)\) \(\chi_{3010}(1863,\cdot)\) \(\chi_{3010}(2003,\cdot)\) \(\chi_{3010}(2073,\cdot)\) \(\chi_{3010}(2087,\cdot)\) \(\chi_{3010}(2353,\cdot)\) \(\chi_{3010}(2423,\cdot)\) \(\chi_{3010}(2507,\cdot)\) \(\chi_{3010}(2577,\cdot)\) \(\chi_{3010}(2633,\cdot)\) \(\chi_{3010}(2647,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1807,1291,1121)\) → \((i,1,e\left(\frac{16}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 3010 }(2507, a) \) \(-1\)\(1\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{11}{84}\right)\)\(e\left(\frac{17}{84}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{79}{84}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{19}{21}\right)\)
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)
 
\( \chi_{ 3010 }(2507,a) \;\) at \(\;a = \) e.g. 2