from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7360, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,0,11,40]))
pari: [g,chi] = znchar(Mod(127,7360))
Basic properties
| Modulus: | \(7360\) | |
| Conductor: | \(460\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{460}(127,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7360.dx
\(\chi_{7360}(127,\cdot)\) \(\chi_{7360}(703,\cdot)\) \(\chi_{7360}(767,\cdot)\) \(\chi_{7360}(1087,\cdot)\) \(\chi_{7360}(1343,\cdot)\) \(\chi_{7360}(1407,\cdot)\) \(\chi_{7360}(1727,\cdot)\) \(\chi_{7360}(2303,\cdot)\) \(\chi_{7360}(3583,\cdot)\) \(\chi_{7360}(3647,\cdot)\) \(\chi_{7360}(3903,\cdot)\) \(\chi_{7360}(4287,\cdot)\) \(\chi_{7360}(4543,\cdot)\) \(\chi_{7360}(5183,\cdot)\) \(\chi_{7360}(5247,\cdot)\) \(\chi_{7360}(5503,\cdot)\) \(\chi_{7360}(5823,\cdot)\) \(\chi_{7360}(6143,\cdot)\) \(\chi_{7360}(6527,\cdot)\) \(\chi_{7360}(6847,\cdot)\)
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
\((1151,5061,4417,6721)\) → \((-1,1,i,e\left(\frac{10}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 7360 }(127, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) |
sage: chi.jacobi_sum(n)