from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1710, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([12,27,14]))
pari: [g,chi] = znchar(Mod(1363,1710))
Basic properties
| Modulus: | \(1710\) | |
| Conductor: | \(855\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{855}(508,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1710.dq
\(\chi_{1710}(193,\cdot)\) \(\chi_{1710}(337,\cdot)\) \(\chi_{1710}(547,\cdot)\) \(\chi_{1710}(553,\cdot)\) \(\chi_{1710}(583,\cdot)\) \(\chi_{1710}(637,\cdot)\) \(\chi_{1710}(877,\cdot)\) \(\chi_{1710}(1237,\cdot)\) \(\chi_{1710}(1267,\cdot)\) \(\chi_{1710}(1363,\cdot)\) \(\chi_{1710}(1573,\cdot)\) \(\chi_{1710}(1663,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.36.17849776228866488737715206984999102954438314099226129130939288096733391284942626953125.1 |
Values on generators
\((191,1027,1351)\) → \((e\left(\frac{1}{3}\right),-i,e\left(\frac{7}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1710 }(1363, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(-1\) | \(i\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) |
sage: chi.jacobi_sum(n)