![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2420, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,0,7]))
![Copy content]()
gp:[g,chi] = znchar(Mod(491, 2420))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2420.491");
| Modulus: | \(2420\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(484\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(110\) |
![Copy content]()
sage:chi.multiplicative_order()
![Copy content]()
gp:charorder(g,chi)
![Copy content]()
magma:Order(chi);
|
| Real: | no |
![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 \(\chi_{484}(7,\cdot)\) |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
gp:#znconreyconductor(g,chi)==1
![Copy content]()
magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{2420}(51,\cdot)\)
\(\chi_{2420}(151,\cdot)\)
\(\chi_{2420}(171,\cdot)\)
\(\chi_{2420}(211,\cdot)\)
\(\chi_{2420}(271,\cdot)\)
\(\chi_{2420}(371,\cdot)\)
\(\chi_{2420}(391,\cdot)\)
\(\chi_{2420}(431,\cdot)\)
\(\chi_{2420}(491,\cdot)\)
\(\chi_{2420}(591,\cdot)\)
\(\chi_{2420}(611,\cdot)\)
\(\chi_{2420}(651,\cdot)\)
\(\chi_{2420}(711,\cdot)\)
\(\chi_{2420}(811,\cdot)\)
\(\chi_{2420}(831,\cdot)\)
\(\chi_{2420}(871,\cdot)\)
\(\chi_{2420}(931,\cdot)\)
\(\chi_{2420}(1031,\cdot)\)
\(\chi_{2420}(1051,\cdot)\)
\(\chi_{2420}(1091,\cdot)\)
\(\chi_{2420}(1151,\cdot)\)
\(\chi_{2420}(1251,\cdot)\)
\(\chi_{2420}(1271,\cdot)\)
\(\chi_{2420}(1311,\cdot)\)
\(\chi_{2420}(1471,\cdot)\)
\(\chi_{2420}(1491,\cdot)\)
\(\chi_{2420}(1531,\cdot)\)
\(\chi_{2420}(1591,\cdot)\)
\(\chi_{2420}(1711,\cdot)\)
\(\chi_{2420}(1751,\cdot)\)
...
![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 };
| Field of values: |
$\Q(\zeta_{55})$ |
![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 110 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((1211,1937,2301)\) → \((-1,1,e\left(\frac{7}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 2420 }(491, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{110}\right)\) |
![Copy content]()
sage:chi(x) # x integer
![Copy content]()
gp:chareval(g,chi,x) \\ x integer, value in Q/Z
![Copy content]()
magma:chi(x)
