from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5415, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,0,10]))
pari: [g,chi] = znchar(Mod(286,5415))
Basic properties
| Modulus: | \(5415\) | |
| Conductor: | \(361\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(19\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{361}(286,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5415.bg
\(\chi_{5415}(286,\cdot)\) \(\chi_{5415}(571,\cdot)\) \(\chi_{5415}(856,\cdot)\) \(\chi_{5415}(1141,\cdot)\) \(\chi_{5415}(1426,\cdot)\) \(\chi_{5415}(1711,\cdot)\) \(\chi_{5415}(1996,\cdot)\) \(\chi_{5415}(2281,\cdot)\) \(\chi_{5415}(2566,\cdot)\) \(\chi_{5415}(2851,\cdot)\) \(\chi_{5415}(3136,\cdot)\) \(\chi_{5415}(3421,\cdot)\) \(\chi_{5415}(3706,\cdot)\) \(\chi_{5415}(3991,\cdot)\) \(\chi_{5415}(4276,\cdot)\) \(\chi_{5415}(4561,\cdot)\) \(\chi_{5415}(4846,\cdot)\) \(\chi_{5415}(5131,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
| Fixed field: | 19.19.10842505080063916320800450434338728415281531281.1 |
Values on generators
\((3611,2167,5056)\) → \((1,1,e\left(\frac{5}{19}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
| \( \chi_{ 5415 }(286, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) |
sage: chi.jacobi_sum(n)