from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2583, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,50,21]))
pari: [g,chi] = znchar(Mod(964,2583))
Basic properties
| Modulus: | \(2583\) | |
| Conductor: | \(287\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{287}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2583.fd
\(\chi_{2583}(964,\cdot)\) \(\chi_{2583}(1027,\cdot)\) \(\chi_{2583}(1153,\cdot)\) \(\chi_{2583}(1279,\cdot)\) \(\chi_{2583}(1333,\cdot)\) \(\chi_{2583}(1396,\cdot)\) \(\chi_{2583}(1468,\cdot)\) \(\chi_{2583}(1522,\cdot)\) \(\chi_{2583}(1594,\cdot)\) \(\chi_{2583}(1648,\cdot)\) \(\chi_{2583}(1720,\cdot)\) \(\chi_{2583}(1783,\cdot)\) \(\chi_{2583}(1837,\cdot)\) \(\chi_{2583}(1963,\cdot)\) \(\chi_{2583}(2089,\cdot)\) \(\chi_{2583}(2152,\cdot)\)
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
\((2297,2215,1072)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{7}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 2583 }(964, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) |
sage: chi.jacobi_sum(n)