![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(836352, base_ring=CyclotomicField(1056))
M = H._module
chi = DirichletCharacter(H, M([0,297,176,192]))
![Copy content]()
gp:[g,chi] = znchar(Mod(38105, 836352))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("836352.38105");
| Modulus: | \(836352\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(139392\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(1056\) |
![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 \(\chi_{139392}(4709,\cdot)\) |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
gp:#znconreyconductor(g,chi)==1
![Copy content]()
magma:IsPrimitive(chi);
|
| Minimal: | no |
| Parity: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{836352}(89,\cdot)\)
\(\chi_{836352}(3257,\cdot)\)
\(\chi_{836352}(8009,\cdot)\)
\(\chi_{836352}(9593,\cdot)\)
\(\chi_{836352}(12761,\cdot)\)
\(\chi_{836352}(14345,\cdot)\)
\(\chi_{836352}(17513,\cdot)\)
\(\chi_{836352}(19097,\cdot)\)
\(\chi_{836352}(23849,\cdot)\)
\(\chi_{836352}(27017,\cdot)\)
\(\chi_{836352}(28601,\cdot)\)
\(\chi_{836352}(31769,\cdot)\)
\(\chi_{836352}(33353,\cdot)\)
\(\chi_{836352}(36521,\cdot)\)
\(\chi_{836352}(38105,\cdot)\)
\(\chi_{836352}(41273,\cdot)\)
\(\chi_{836352}(42857,\cdot)\)
\(\chi_{836352}(46025,\cdot)\)
\(\chi_{836352}(47609,\cdot)\)
\(\chi_{836352}(50777,\cdot)\)
\(\chi_{836352}(52361,\cdot)\)
\(\chi_{836352}(55529,\cdot)\)
\(\chi_{836352}(60281,\cdot)\)
\(\chi_{836352}(61865,\cdot)\)
\(\chi_{836352}(65033,\cdot)\)
\(\chi_{836352}(66617,\cdot)\)
\(\chi_{836352}(69785,\cdot)\)
\(\chi_{836352}(71369,\cdot)\)
\(\chi_{836352}(76121,\cdot)\)
\(\chi_{836352}(79289,\cdot)\)
...
![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: |
$\Q(\zeta_{1056})$ |
![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 1056 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((137215,561925,123905,781057)\) → \((1,e\left(\frac{9}{32}\right),e\left(\frac{1}{6}\right),e\left(\frac{2}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 836352 }(38105, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{601}{1056}\right)\) | \(e\left(\frac{397}{528}\right)\) | \(e\left(\frac{967}{1056}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{197}{352}\right)\) | \(e\left(\frac{263}{528}\right)\) | \(e\left(\frac{73}{528}\right)\) | \(e\left(\frac{899}{1056}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{113}{352}\right)\) |
![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)
