from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7360, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,22,32]))
pari: [g,chi] = znchar(Mod(49,7360))
Basic properties
| Modulus: | \(7360\) | |
| Conductor: | \(1840\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1840}(1429,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7360.dq
\(\chi_{7360}(49,\cdot)\) \(\chi_{7360}(209,\cdot)\) \(\chi_{7360}(1329,\cdot)\) \(\chi_{7360}(1649,\cdot)\) \(\chi_{7360}(2129,\cdot)\) \(\chi_{7360}(2289,\cdot)\) \(\chi_{7360}(2769,\cdot)\) \(\chi_{7360}(2929,\cdot)\) \(\chi_{7360}(3249,\cdot)\) \(\chi_{7360}(3569,\cdot)\) \(\chi_{7360}(3729,\cdot)\) \(\chi_{7360}(3889,\cdot)\) \(\chi_{7360}(5009,\cdot)\) \(\chi_{7360}(5329,\cdot)\) \(\chi_{7360}(5809,\cdot)\) \(\chi_{7360}(5969,\cdot)\) \(\chi_{7360}(6449,\cdot)\) \(\chi_{7360}(6609,\cdot)\) \(\chi_{7360}(6929,\cdot)\) \(\chi_{7360}(7249,\cdot)\)
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
\((1151,5061,4417,6721)\) → \((1,i,-1,e\left(\frac{8}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 7360 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) |
sage: chi.jacobi_sum(n)