from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2790, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,45,52]))
pari: [g,chi] = znchar(Mod(173,2790))
Basic properties
| Modulus: | \(2790\) | |
| Conductor: | \(1395\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1395}(173,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2790.eo
\(\chi_{2790}(173,\cdot)\) \(\chi_{2790}(227,\cdot)\) \(\chi_{2790}(293,\cdot)\) \(\chi_{2790}(443,\cdot)\) \(\chi_{2790}(617,\cdot)\) \(\chi_{2790}(857,\cdot)\) \(\chi_{2790}(887,\cdot)\) \(\chi_{2790}(1247,\cdot)\) \(\chi_{2790}(1343,\cdot)\) \(\chi_{2790}(1733,\cdot)\) \(\chi_{2790}(1847,\cdot)\) \(\chi_{2790}(1967,\cdot)\) \(\chi_{2790}(1973,\cdot)\) \(\chi_{2790}(2003,\cdot)\) \(\chi_{2790}(2117,\cdot)\) \(\chi_{2790}(2363,\cdot)\)
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
\((2171,1117,1801)\) → \((e\left(\frac{1}{6}\right),-i,e\left(\frac{13}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2790 }(173, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) |
sage: chi.jacobi_sum(n)