from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5520, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,0,33,28]))
pari: [g,chi] = znchar(Mod(13,5520))
Basic properties
| Modulus: | \(5520\) | |
| 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 \(\chi_{1840}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5520.fj
\(\chi_{5520}(13,\cdot)\) \(\chi_{5520}(997,\cdot)\) \(\chi_{5520}(1237,\cdot)\) \(\chi_{5520}(1453,\cdot)\) \(\chi_{5520}(1957,\cdot)\) \(\chi_{5520}(2197,\cdot)\) \(\chi_{5520}(2653,\cdot)\) \(\chi_{5520}(2677,\cdot)\) \(\chi_{5520}(2893,\cdot)\) \(\chi_{5520}(3157,\cdot)\) \(\chi_{5520}(3397,\cdot)\) \(\chi_{5520}(3613,\cdot)\) \(\chi_{5520}(3637,\cdot)\) \(\chi_{5520}(3853,\cdot)\) \(\chi_{5520}(3877,\cdot)\) \(\chi_{5520}(4333,\cdot)\) \(\chi_{5520}(4813,\cdot)\) \(\chi_{5520}(5053,\cdot)\) \(\chi_{5520}(5293,\cdot)\) \(\chi_{5520}(5317,\cdot)\)
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
\((4831,1381,1841,4417,1201)\) → \((1,-i,1,-i,e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5520 }(13, a) \) | \(-1\) | \(1\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) |
sage: chi.jacobi_sum(n)