from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8400, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,0,57,20]))
pari: [g,chi] = znchar(Mod(163,8400))
Basic properties
| Modulus: | \(8400\) | |
| Conductor: | \(2800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2800}(163,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8400.lt
\(\chi_{8400}(163,\cdot)\) \(\chi_{8400}(403,\cdot)\) \(\chi_{8400}(667,\cdot)\) \(\chi_{8400}(2083,\cdot)\) \(\chi_{8400}(2347,\cdot)\) \(\chi_{8400}(2587,\cdot)\) \(\chi_{8400}(3523,\cdot)\) \(\chi_{8400}(3763,\cdot)\) \(\chi_{8400}(4027,\cdot)\) \(\chi_{8400}(4267,\cdot)\) \(\chi_{8400}(5203,\cdot)\) \(\chi_{8400}(5947,\cdot)\) \(\chi_{8400}(6883,\cdot)\) \(\chi_{8400}(7123,\cdot)\) \(\chi_{8400}(7387,\cdot)\) \(\chi_{8400}(7627,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((3151,2101,2801,5377,3601)\) → \((-1,-i,1,e\left(\frac{19}{20}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8400 }(163, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)