from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1470, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,7,20]))
pari: [g,chi] = znchar(Mod(407,1470))
Basic properties
| Modulus: | \(1470\) | |
| Conductor: | \(735\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{735}(407,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1470.bj
\(\chi_{1470}(113,\cdot)\) \(\chi_{1470}(323,\cdot)\) \(\chi_{1470}(407,\cdot)\) \(\chi_{1470}(533,\cdot)\) \(\chi_{1470}(617,\cdot)\) \(\chi_{1470}(743,\cdot)\) \(\chi_{1470}(827,\cdot)\) \(\chi_{1470}(953,\cdot)\) \(\chi_{1470}(1037,\cdot)\) \(\chi_{1470}(1163,\cdot)\) \(\chi_{1470}(1247,\cdot)\) \(\chi_{1470}(1457,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((491,1177,1081)\) → \((-1,i,e\left(\frac{5}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1470 }(407, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(-1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) |
sage: chi.jacobi_sum(n)