from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1350, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,51]))
pari: [g,chi] = znchar(Mod(1097,1350))
Basic properties
| Modulus: | \(1350\) | |
| Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{225}(122,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1350.bd
\(\chi_{1350}(17,\cdot)\) \(\chi_{1350}(197,\cdot)\) \(\chi_{1350}(233,\cdot)\) \(\chi_{1350}(287,\cdot)\) \(\chi_{1350}(413,\cdot)\) \(\chi_{1350}(467,\cdot)\) \(\chi_{1350}(503,\cdot)\) \(\chi_{1350}(683,\cdot)\) \(\chi_{1350}(737,\cdot)\) \(\chi_{1350}(773,\cdot)\) \(\chi_{1350}(827,\cdot)\) \(\chi_{1350}(953,\cdot)\) \(\chi_{1350}(1097,\cdot)\) \(\chi_{1350}(1223,\cdot)\) \(\chi_{1350}(1277,\cdot)\) \(\chi_{1350}(1313,\cdot)\)
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
\((1001,1027)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{17}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1350 }(1097, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) |
sage: chi.jacobi_sum(n)