from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1710, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([4,9,10]))
pari: [g,chi] = znchar(Mod(373,1710))
Basic properties
| Modulus: | \(1710\) | |
| Conductor: | \(855\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{855}(373,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1710.cc
\(\chi_{1710}(373,\cdot)\) \(\chi_{1710}(673,\cdot)\) \(\chi_{1710}(1057,\cdot)\) \(\chi_{1710}(1357,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
| Fixed field: | 12.12.515473239515768985392578125.2 |
Values on generators
\((191,1027,1351)\) → \((e\left(\frac{1}{3}\right),-i,e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1710 }(373, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(-1\) | \(e\left(\frac{11}{12}\right)\) |
sage: chi.jacobi_sum(n)