![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2675, base_ring=CyclotomicField(530))
M = H._module
chi = DirichletCharacter(H, M([106,35]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1091, 2675))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2675.1091");
| Modulus: | \(2675\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(2675\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(530\) |
![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: | yes |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
gp:#znconreyconductor(g,chi)==1
![Copy content]()
magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{2675}(6,\cdot)\)
\(\chi_{2675}(21,\cdot)\)
\(\chi_{2675}(31,\cdot)\)
\(\chi_{2675}(46,\cdot)\)
\(\chi_{2675}(66,\cdot)\)
\(\chi_{2675}(71,\cdot)\)
\(\chi_{2675}(91,\cdot)\)
\(\chi_{2675}(96,\cdot)\)
\(\chi_{2675}(131,\cdot)\)
\(\chi_{2675}(161,\cdot)\)
\(\chi_{2675}(166,\cdot)\)
\(\chi_{2675}(181,\cdot)\)
\(\chi_{2675}(191,\cdot)\)
\(\chi_{2675}(211,\cdot)\)
\(\chi_{2675}(216,\cdot)\)
\(\chi_{2675}(221,\cdot)\)
\(\chi_{2675}(231,\cdot)\)
\(\chi_{2675}(236,\cdot)\)
\(\chi_{2675}(246,\cdot)\)
\(\chi_{2675}(281,\cdot)\)
\(\chi_{2675}(286,\cdot)\)
\(\chi_{2675}(291,\cdot)\)
\(\chi_{2675}(296,\cdot)\)
\(\chi_{2675}(311,\cdot)\)
\(\chi_{2675}(336,\cdot)\)
\(\chi_{2675}(341,\cdot)\)
\(\chi_{2675}(366,\cdot)\)
\(\chi_{2675}(371,\cdot)\)
\(\chi_{2675}(381,\cdot)\)
\(\chi_{2675}(386,\cdot)\)
...
![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: |
$\Q(\zeta_{265})$ |
![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 530 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((1927,751)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{7}{106}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 2675 }(1091, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{141}{530}\right)\) | \(e\left(\frac{6}{265}\right)\) | \(e\left(\frac{141}{265}\right)\) | \(e\left(\frac{153}{530}\right)\) | \(e\left(\frac{89}{106}\right)\) | \(e\left(\frac{423}{530}\right)\) | \(e\left(\frac{12}{265}\right)\) | \(e\left(\frac{173}{265}\right)\) | \(e\left(\frac{147}{265}\right)\) | \(e\left(\frac{192}{265}\right)\) |
![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)
