from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1209, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,15,19]))
pari: [g,chi] = znchar(Mod(818,1209))
Basic properties
| Modulus: | \(1209\) | |
| Conductor: | \(1209\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | 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 1209.cy
\(\chi_{1209}(272,\cdot)\) \(\chi_{1209}(389,\cdot)\) \(\chi_{1209}(623,\cdot)\) \(\chi_{1209}(662,\cdot)\) \(\chi_{1209}(818,\cdot)\) \(\chi_{1209}(974,\cdot)\) \(\chi_{1209}(1013,\cdot)\) \(\chi_{1209}(1169,\cdot)\)
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: | Number field defined by a degree 30 polynomial |
Values on generators
\((404,652,313)\) → \((-1,-1,e\left(\frac{19}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1209 }(818, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) |
sage: chi.jacobi_sum(n)