from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1476, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,10,33]))
pari: [g,chi] = znchar(Mod(497,1476))
Basic properties
| Modulus: | \(1476\) | |
| Conductor: | \(369\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{369}(128,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1476.cg
\(\chi_{1476}(5,\cdot)\) \(\chi_{1476}(77,\cdot)\) \(\chi_{1476}(185,\cdot)\) \(\chi_{1476}(389,\cdot)\) \(\chi_{1476}(497,\cdot)\) \(\chi_{1476}(569,\cdot)\) \(\chi_{1476}(617,\cdot)\) \(\chi_{1476}(677,\cdot)\) \(\chi_{1476}(689,\cdot)\) \(\chi_{1476}(869,\cdot)\) \(\chi_{1476}(941,\cdot)\) \(\chi_{1476}(1109,\cdot)\) \(\chi_{1476}(1181,\cdot)\) \(\chi_{1476}(1361,\cdot)\) \(\chi_{1476}(1373,\cdot)\) \(\chi_{1476}(1433,\cdot)\)
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
\((739,821,1441)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{11}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1476 }(497, a) \) | \(-1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) |
sage: chi.jacobi_sum(n)