Properties

Label 676000.bva
Modulus $676000$
Conductor $676000$
Order $7800$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character orbit
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(676000, base_ring=CyclotomicField(7800)) M = H._module chi = DirichletCharacter(H, M([3900,2925,546,6200])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(3, 676000)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("676000.3"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(676000\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(676000\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(7800\)
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);
 

Related number fields

Field of values: $\Q(\zeta_{7800})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 7800 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

First 5 of 1920 characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(7\) \(9\) \(11\) \(17\) \(19\) \(21\) \(23\) \(27\) \(29\)
\(\chi_{676000}(3,\cdot)\) \(1\) \(1\) \(e\left(\frac{5297}{7800}\right)\) \(e\left(\frac{49}{195}\right)\) \(e\left(\frac{1397}{3900}\right)\) \(e\left(\frac{4421}{7800}\right)\) \(e\left(\frac{2579}{3900}\right)\) \(e\left(\frac{31}{600}\right)\) \(e\left(\frac{2419}{2600}\right)\) \(e\left(\frac{19}{75}\right)\) \(e\left(\frac{97}{2600}\right)\) \(e\left(\frac{2027}{7800}\right)\)
\(\chi_{676000}(347,\cdot)\) \(1\) \(1\) \(e\left(\frac{4627}{7800}\right)\) \(e\left(\frac{59}{195}\right)\) \(e\left(\frac{727}{3900}\right)\) \(e\left(\frac{7711}{7800}\right)\) \(e\left(\frac{2389}{3900}\right)\) \(e\left(\frac{221}{600}\right)\) \(e\left(\frac{2329}{2600}\right)\) \(e\left(\frac{29}{75}\right)\) \(e\left(\frac{2027}{2600}\right)\) \(e\left(\frac{7057}{7800}\right)\)
\(\chi_{676000}(1147,\cdot)\) \(1\) \(1\) \(e\left(\frac{7667}{7800}\right)\) \(e\left(\frac{34}{195}\right)\) \(e\left(\frac{3767}{3900}\right)\) \(e\left(\frac{1631}{7800}\right)\) \(e\left(\frac{2669}{3900}\right)\) \(e\left(\frac{541}{600}\right)\) \(e\left(\frac{409}{2600}\right)\) \(e\left(\frac{34}{75}\right)\) \(e\left(\frac{2467}{2600}\right)\) \(e\left(\frac{1697}{7800}\right)\)
\(\chi_{676000}(1283,\cdot)\) \(1\) \(1\) \(e\left(\frac{2593}{7800}\right)\) \(e\left(\frac{101}{195}\right)\) \(e\left(\frac{2593}{3900}\right)\) \(e\left(\frac{5149}{7800}\right)\) \(e\left(\frac{3151}{3900}\right)\) \(e\left(\frac{239}{600}\right)\) \(e\left(\frac{2211}{2600}\right)\) \(e\left(\frac{11}{75}\right)\) \(e\left(\frac{2593}{2600}\right)\) \(e\left(\frac{5563}{7800}\right)\)
\(\chi_{676000}(1387,\cdot)\) \(1\) \(1\) \(e\left(\frac{1639}{7800}\right)\) \(e\left(\frac{188}{195}\right)\) \(e\left(\frac{1639}{3900}\right)\) \(e\left(\frac{427}{7800}\right)\) \(e\left(\frac{2473}{3900}\right)\) \(e\left(\frac{497}{600}\right)\) \(e\left(\frac{453}{2600}\right)\) \(e\left(\frac{53}{75}\right)\) \(e\left(\frac{1639}{2600}\right)\) \(e\left(\frac{1549}{7800}\right)\)