from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4592, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,10,51]))
pari: [g,chi] = znchar(Mod(143,4592))
Basic properties
| Modulus: | \(4592\) | |
| Conductor: | \(1148\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1148}(143,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4592.ht
\(\chi_{4592}(143,\cdot)\) \(\chi_{4592}(159,\cdot)\) \(\chi_{4592}(367,\cdot)\) \(\chi_{4592}(607,\cdot)\) \(\chi_{4592}(815,\cdot)\) \(\chi_{4592}(1263,\cdot)\) \(\chi_{4592}(1279,\cdot)\) \(\chi_{4592}(1727,\cdot)\) \(\chi_{4592}(1935,\cdot)\) \(\chi_{4592}(2175,\cdot)\) \(\chi_{4592}(2383,\cdot)\) \(\chi_{4592}(2399,\cdot)\) \(\chi_{4592}(2831,\cdot)\) \(\chi_{4592}(3055,\cdot)\) \(\chi_{4592}(4079,\cdot)\) \(\chi_{4592}(4303,\cdot)\)
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
\((575,3445,3937,785)\) → \((-1,1,e\left(\frac{1}{6}\right),e\left(\frac{17}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 4592 }(143, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) |
sage: chi.jacobi_sum(n)