from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1380, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,22,11,36]))
pari: [g,chi] = znchar(Mod(167,1380))
Basic properties
| Modulus: | \(1380\) | |
| Conductor: | \(1380\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1380.bp
\(\chi_{1380}(167,\cdot)\) \(\chi_{1380}(347,\cdot)\) \(\chi_{1380}(407,\cdot)\) \(\chi_{1380}(443,\cdot)\) \(\chi_{1380}(587,\cdot)\) \(\chi_{1380}(623,\cdot)\) \(\chi_{1380}(647,\cdot)\) \(\chi_{1380}(683,\cdot)\) \(\chi_{1380}(767,\cdot)\) \(\chi_{1380}(863,\cdot)\) \(\chi_{1380}(887,\cdot)\) \(\chi_{1380}(923,\cdot)\) \(\chi_{1380}(947,\cdot)\) \(\chi_{1380}(1007,\cdot)\) \(\chi_{1380}(1043,\cdot)\) \(\chi_{1380}(1067,\cdot)\) \(\chi_{1380}(1163,\cdot)\) \(\chi_{1380}(1223,\cdot)\) \(\chi_{1380}(1283,\cdot)\) \(\chi_{1380}(1343,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((691,461,277,1201)\) → \((-1,-1,i,e\left(\frac{9}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1380 }(167, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{15}{44}\right)\) |
sage: chi.jacobi_sum(n)