![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(17675, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([75,50,141]))
![Copy content]()
gp:[g,chi] = znchar(Mod(12049, 17675))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17675.12049");
| Modulus: | \(17675\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(3535\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(150\) |
![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 \(\chi_{3535}(1444,\cdot)\) |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
gp:#znconreyconductor(g,chi)==1
![Copy content]()
magma:IsPrimitive(chi);
|
| Minimal: | no |
| Parity: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{17675}(249,\cdot)\)
\(\chi_{17675}(324,\cdot)\)
\(\chi_{17675}(424,\cdot)\)
\(\chi_{17675}(1124,\cdot)\)
\(\chi_{17675}(1649,\cdot)\)
\(\chi_{17675}(2774,\cdot)\)
\(\chi_{17675}(2949,\cdot)\)
\(\chi_{17675}(3649,\cdot)\)
\(\chi_{17675}(4174,\cdot)\)
\(\chi_{17675}(5499,\cdot)\)
\(\chi_{17675}(6549,\cdot)\)
\(\chi_{17675}(7074,\cdot)\)
\(\chi_{17675}(8024,\cdot)\)
\(\chi_{17675}(8649,\cdot)\)
\(\chi_{17675}(9074,\cdot)\)
\(\chi_{17675}(9524,\cdot)\)
\(\chi_{17675}(9599,\cdot)\)
\(\chi_{17675}(9874,\cdot)\)
\(\chi_{17675}(10224,\cdot)\)
\(\chi_{17675}(10574,\cdot)\)
\(\chi_{17675}(10749,\cdot)\)
\(\chi_{17675}(11174,\cdot)\)
\(\chi_{17675}(11624,\cdot)\)
\(\chi_{17675}(12049,\cdot)\)
\(\chi_{17675}(12399,\cdot)\)
\(\chi_{17675}(12499,\cdot)\)
\(\chi_{17675}(12674,\cdot)\)
\(\chi_{17675}(12749,\cdot)\)
\(\chi_{17675}(12849,\cdot)\)
\(\chi_{17675}(13024,\cdot)\)
...
![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 };
\((12727,15151,15051)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{47}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 17675 }(12049, a) \) |
\(1\) | \(1\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{83}{150}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{32}{75}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
