from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1520, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,9,14]))
pari: [g,chi] = znchar(Mod(1477,1520))
Basic properties
| Modulus: | \(1520\) | |
| Conductor: | \(1520\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1520.ed
\(\chi_{1520}(13,\cdot)\) \(\chi_{1520}(117,\cdot)\) \(\chi_{1520}(173,\cdot)\) \(\chi_{1520}(333,\cdot)\) \(\chi_{1520}(357,\cdot)\) \(\chi_{1520}(413,\cdot)\) \(\chi_{1520}(573,\cdot)\) \(\chi_{1520}(813,\cdot)\) \(\chi_{1520}(1077,\cdot)\) \(\chi_{1520}(1237,\cdot)\) \(\chi_{1520}(1397,\cdot)\) \(\chi_{1520}(1477,\cdot)\)
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: | Number field defined by a degree 36 polynomial |
Values on generators
\((191,1141,1217,401)\) → \((1,i,i,e\left(\frac{7}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1520 }(1477, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{31}{36}\right)\) |
sage: chi.jacobi_sum(n)