from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,11,18]))
pari: [g,chi] = znchar(Mod(31,2415))
Basic properties
| Modulus: | \(2415\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{161}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2415.di
\(\chi_{2415}(31,\cdot)\) \(\chi_{2415}(271,\cdot)\) \(\chi_{2415}(376,\cdot)\) \(\chi_{2415}(556,\cdot)\) \(\chi_{2415}(901,\cdot)\) \(\chi_{2415}(1186,\cdot)\) \(\chi_{2415}(1291,\cdot)\) \(\chi_{2415}(1396,\cdot)\) \(\chi_{2415}(1501,\cdot)\) \(\chi_{2415}(1531,\cdot)\) \(\chi_{2415}(1636,\cdot)\) \(\chi_{2415}(1711,\cdot)\) \(\chi_{2415}(1741,\cdot)\) \(\chi_{2415}(1846,\cdot)\) \(\chi_{2415}(1921,\cdot)\) \(\chi_{2415}(2026,\cdot)\) \(\chi_{2415}(2056,\cdot)\) \(\chi_{2415}(2266,\cdot)\) \(\chi_{2415}(2341,\cdot)\) \(\chi_{2415}(2371,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((806,967,346,1891)\) → \((1,1,e\left(\frac{1}{6}\right),e\left(\frac{3}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(31, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(1\) | \(e\left(\frac{13}{66}\right)\) |
sage: chi.jacobi_sum(n)