from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2450, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([39,40]))
pari: [g,chi] = znchar(Mod(67,2450))
Basic properties
| Modulus: | \(2450\) | |
| Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{175}(67,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2450.bh
\(\chi_{2450}(67,\cdot)\) \(\chi_{2450}(177,\cdot)\) \(\chi_{2450}(263,\cdot)\) \(\chi_{2450}(373,\cdot)\) \(\chi_{2450}(667,\cdot)\) \(\chi_{2450}(753,\cdot)\) \(\chi_{2450}(863,\cdot)\) \(\chi_{2450}(1047,\cdot)\) \(\chi_{2450}(1353,\cdot)\) \(\chi_{2450}(1537,\cdot)\) \(\chi_{2450}(1647,\cdot)\) \(\chi_{2450}(1733,\cdot)\) \(\chi_{2450}(2027,\cdot)\) \(\chi_{2450}(2137,\cdot)\) \(\chi_{2450}(2223,\cdot)\) \(\chi_{2450}(2333,\cdot)\)
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
\((1177,101)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 2450 }(67, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) |
sage: chi.jacobi_sum(n)