from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5520, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,22,33,8]))
pari: [g,chi] = znchar(Mod(1913,5520))
Basic properties
| Modulus: | \(5520\) | |
| Conductor: | \(2760\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2760}(533,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5520.es
\(\chi_{5520}(233,\cdot)\) \(\chi_{5520}(377,\cdot)\) \(\chi_{5520}(473,\cdot)\) \(\chi_{5520}(857,\cdot)\) \(\chi_{5520}(1097,\cdot)\) \(\chi_{5520}(1337,\cdot)\) \(\chi_{5520}(1577,\cdot)\) \(\chi_{5520}(1913,\cdot)\) \(\chi_{5520}(3017,\cdot)\) \(\chi_{5520}(3113,\cdot)\) \(\chi_{5520}(3353,\cdot)\) \(\chi_{5520}(4073,\cdot)\) \(\chi_{5520}(4217,\cdot)\) \(\chi_{5520}(4313,\cdot)\) \(\chi_{5520}(4457,\cdot)\) \(\chi_{5520}(4793,\cdot)\) \(\chi_{5520}(5177,\cdot)\) \(\chi_{5520}(5273,\cdot)\) \(\chi_{5520}(5417,\cdot)\) \(\chi_{5520}(5513,\cdot)\)
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
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5520 }(1913, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) |
sage: chi.jacobi_sum(n)