from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1155, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,0,25,27]))
pari: [g,chi] = znchar(Mod(61,1155))
Basic properties
| Modulus: | \(1155\) | |
| Conductor: | \(77\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{77}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1155.cy
\(\chi_{1155}(61,\cdot)\) \(\chi_{1155}(271,\cdot)\) \(\chi_{1155}(376,\cdot)\) \(\chi_{1155}(481,\cdot)\) \(\chi_{1155}(556,\cdot)\) \(\chi_{1155}(766,\cdot)\) \(\chi_{1155}(871,\cdot)\) \(\chi_{1155}(976,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | \(\Q(\zeta_{77})^+\) |
Values on generators
\((386,232,661,211)\) → \((1,1,e\left(\frac{5}{6}\right),e\left(\frac{9}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
| \( \chi_{ 1155 }(61, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)