from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1232, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,50,36]))
pari: [g,chi] = znchar(Mod(75,1232))
Basic properties
| Modulus: | \(1232\) | |
| Conductor: | \(1232\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1232.dp
\(\chi_{1232}(3,\cdot)\) \(\chi_{1232}(59,\cdot)\) \(\chi_{1232}(75,\cdot)\) \(\chi_{1232}(115,\cdot)\) \(\chi_{1232}(339,\cdot)\) \(\chi_{1232}(355,\cdot)\) \(\chi_{1232}(411,\cdot)\) \(\chi_{1232}(467,\cdot)\) \(\chi_{1232}(619,\cdot)\) \(\chi_{1232}(675,\cdot)\) \(\chi_{1232}(691,\cdot)\) \(\chi_{1232}(731,\cdot)\) \(\chi_{1232}(955,\cdot)\) \(\chi_{1232}(971,\cdot)\) \(\chi_{1232}(1027,\cdot)\) \(\chi_{1232}(1083,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((463,309,353,673)\) → \((-1,i,e\left(\frac{5}{6}\right),e\left(\frac{3}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 1232 }(75, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) |
sage: chi.jacobi_sum(n)