![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1728, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([72,63,8]))
![Copy content]()
gp:[g,chi] = znchar(Mod(83, 1728))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1728.83");
| Modulus: | \(1728\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(1728\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(144\) |
![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: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{1728}(11,\cdot)\)
\(\chi_{1728}(59,\cdot)\)
\(\chi_{1728}(83,\cdot)\)
\(\chi_{1728}(131,\cdot)\)
\(\chi_{1728}(155,\cdot)\)
\(\chi_{1728}(203,\cdot)\)
\(\chi_{1728}(227,\cdot)\)
\(\chi_{1728}(275,\cdot)\)
\(\chi_{1728}(299,\cdot)\)
\(\chi_{1728}(347,\cdot)\)
\(\chi_{1728}(371,\cdot)\)
\(\chi_{1728}(419,\cdot)\)
\(\chi_{1728}(443,\cdot)\)
\(\chi_{1728}(491,\cdot)\)
\(\chi_{1728}(515,\cdot)\)
\(\chi_{1728}(563,\cdot)\)
\(\chi_{1728}(587,\cdot)\)
\(\chi_{1728}(635,\cdot)\)
\(\chi_{1728}(659,\cdot)\)
\(\chi_{1728}(707,\cdot)\)
\(\chi_{1728}(731,\cdot)\)
\(\chi_{1728}(779,\cdot)\)
\(\chi_{1728}(803,\cdot)\)
\(\chi_{1728}(851,\cdot)\)
\(\chi_{1728}(875,\cdot)\)
\(\chi_{1728}(923,\cdot)\)
\(\chi_{1728}(947,\cdot)\)
\(\chi_{1728}(995,\cdot)\)
\(\chi_{1728}(1019,\cdot)\)
\(\chi_{1728}(1067,\cdot)\)
...
![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: |
$\Q(\zeta_{144})$ |
![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 144 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((703,325,1217)\) → \((-1,e\left(\frac{7}{16}\right),e\left(\frac{1}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1728 }(83, a) \) |
\(1\) | \(1\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{59}{144}\right)\) | \(e\left(\frac{1}{144}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{125}{144}\right)\) | \(e\left(\frac{1}{9}\right)\) |
![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)
