from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5550, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,27,20]))
pari: [g,chi] = znchar(Mod(137,5550))
Basic properties
| Modulus: | \(5550\) | |
| Conductor: | \(2775\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2775}(137,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5550.ds
\(\chi_{5550}(47,\cdot)\) \(\chi_{5550}(137,\cdot)\) \(\chi_{5550}(713,\cdot)\) \(\chi_{5550}(803,\cdot)\) \(\chi_{5550}(1247,\cdot)\) \(\chi_{5550}(1823,\cdot)\) \(\chi_{5550}(1913,\cdot)\) \(\chi_{5550}(2267,\cdot)\) \(\chi_{5550}(2933,\cdot)\) \(\chi_{5550}(3023,\cdot)\) \(\chi_{5550}(3377,\cdot)\) \(\chi_{5550}(3467,\cdot)\) \(\chi_{5550}(4133,\cdot)\) \(\chi_{5550}(4487,\cdot)\) \(\chi_{5550}(4577,\cdot)\) \(\chi_{5550}(5153,\cdot)\)
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
\((3701,1777,2851)\) → \((-1,e\left(\frac{9}{20}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
| \( \chi_{ 5550 }(137, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)