from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5520, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,22,0,34]))
pari: [g,chi] = znchar(Mod(4661,5520))
Basic properties
| Modulus: | \(5520\) | |
| Conductor: | \(1104\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1104}(245,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5520.ei
\(\chi_{5520}(221,\cdot)\) \(\chi_{5520}(341,\cdot)\) \(\chi_{5520}(701,\cdot)\) \(\chi_{5520}(941,\cdot)\) \(\chi_{5520}(1661,\cdot)\) \(\chi_{5520}(1781,\cdot)\) \(\chi_{5520}(1901,\cdot)\) \(\chi_{5520}(2021,\cdot)\) \(\chi_{5520}(2261,\cdot)\) \(\chi_{5520}(2501,\cdot)\) \(\chi_{5520}(2981,\cdot)\) \(\chi_{5520}(3101,\cdot)\) \(\chi_{5520}(3461,\cdot)\) \(\chi_{5520}(3701,\cdot)\) \(\chi_{5520}(4421,\cdot)\) \(\chi_{5520}(4541,\cdot)\) \(\chi_{5520}(4661,\cdot)\) \(\chi_{5520}(4781,\cdot)\) \(\chi_{5520}(5021,\cdot)\) \(\chi_{5520}(5261,\cdot)\)
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
\((4831,1381,1841,4417,1201)\) → \((1,i,-1,1,e\left(\frac{17}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5520 }(4661, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) |
sage: chi.jacobi_sum(n)