![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12045, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([90,45,126,65]))
![Copy content]()
gp:[g,chi] = znchar(Mod(6332, 12045))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12045.6332");
| Modulus: | \(12045\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(12045\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(180\) |
![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);
|
\(\chi_{12045}(413,\cdot)\)
\(\chi_{12045}(1118,\cdot)\)
\(\chi_{12045}(1667,\cdot)\)
\(\chi_{12045}(2063,\cdot)\)
\(\chi_{12045}(2213,\cdot)\)
\(\chi_{12045}(2228,\cdot)\)
\(\chi_{12045}(2603,\cdot)\)
\(\chi_{12045}(2987,\cdot)\)
\(\chi_{12045}(3218,\cdot)\)
\(\chi_{12045}(3308,\cdot)\)
\(\chi_{12045}(3698,\cdot)\)
\(\chi_{12045}(3857,\cdot)\)
\(\chi_{12045}(3977,\cdot)\)
\(\chi_{12045}(4142,\cdot)\)
\(\chi_{12045}(4253,\cdot)\)
\(\chi_{12045}(4418,\cdot)\)
\(\chi_{12045}(4538,\cdot)\)
\(\chi_{12045}(4793,\cdot)\)
\(\chi_{12045}(4952,\cdot)\)
\(\chi_{12045}(5177,\cdot)\)
\(\chi_{12045}(5348,\cdot)\)
\(\chi_{12045}(5408,\cdot)\)
\(\chi_{12045}(5513,\cdot)\)
\(\chi_{12045}(5792,\cdot)\)
\(\chi_{12045}(6047,\cdot)\)
\(\chi_{12045}(6167,\cdot)\)
\(\chi_{12045}(6272,\cdot)\)
\(\chi_{12045}(6332,\cdot)\)
\(\chi_{12045}(6443,\cdot)\)
\(\chi_{12045}(6503,\cdot)\)
...
![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 };
\((4016,9637,2191,8911)\) → \((-1,i,e\left(\frac{7}{10}\right),e\left(\frac{13}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 12045 }(6332, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{61}{180}\right)\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{73}{180}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{31}{36}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
