from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6930, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,10,42]))
pari: [g,chi] = znchar(Mod(73,6930))
Basic properties
| Modulus: | \(6930\) | |
| Conductor: | \(385\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{385}(73,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6930.ie
\(\chi_{6930}(73,\cdot)\) \(\chi_{6930}(523,\cdot)\) \(\chi_{6930}(1207,\cdot)\) \(\chi_{6930}(1333,\cdot)\) \(\chi_{6930}(1657,\cdot)\) \(\chi_{6930}(2593,\cdot)\) \(\chi_{6930}(2917,\cdot)\) \(\chi_{6930}(3043,\cdot)\) \(\chi_{6930}(3097,\cdot)\) \(\chi_{6930}(4177,\cdot)\) \(\chi_{6930}(4303,\cdot)\) \(\chi_{6930}(4483,\cdot)\) \(\chi_{6930}(5563,\cdot)\) \(\chi_{6930}(5617,\cdot)\) \(\chi_{6930}(6067,\cdot)\) \(\chi_{6930}(6877,\cdot)\)
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
\((1541,1387,2971,2521)\) → \((1,-i,e\left(\frac{1}{6}\right),e\left(\frac{7}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 6930 }(73, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(-i\) | \(e\left(\frac{11}{60}\right)\) |
sage: chi.jacobi_sum(n)