from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5520, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,0,33,8]))
pari: [g,chi] = znchar(Mod(73,5520))
Basic properties
| Modulus: | \(5520\) | |
| Conductor: | \(920\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{920}(533,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5520.fo
\(\chi_{5520}(73,\cdot)\) \(\chi_{5520}(1177,\cdot)\) \(\chi_{5520}(1273,\cdot)\) \(\chi_{5520}(1513,\cdot)\) \(\chi_{5520}(2233,\cdot)\) \(\chi_{5520}(2377,\cdot)\) \(\chi_{5520}(2473,\cdot)\) \(\chi_{5520}(2617,\cdot)\) \(\chi_{5520}(2953,\cdot)\) \(\chi_{5520}(3337,\cdot)\) \(\chi_{5520}(3433,\cdot)\) \(\chi_{5520}(3577,\cdot)\) \(\chi_{5520}(3673,\cdot)\) \(\chi_{5520}(3913,\cdot)\) \(\chi_{5520}(4057,\cdot)\) \(\chi_{5520}(4153,\cdot)\) \(\chi_{5520}(4537,\cdot)\) \(\chi_{5520}(4777,\cdot)\) \(\chi_{5520}(5017,\cdot)\) \(\chi_{5520}(5257,\cdot)\)
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
\((4831,1381,1841,4417,1201)\) → \((1,-1,1,-i,e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5520 }(73, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) |
sage: chi.jacobi_sum(n)