from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5520, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,0,0,36]))
pari: [g,chi] = znchar(Mod(4261,5520))
Basic properties
| Modulus: | \(5520\) | |
| Conductor: | \(368\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{368}(213,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5520.fx
\(\chi_{5520}(301,\cdot)\) \(\chi_{5520}(541,\cdot)\) \(\chi_{5520}(901,\cdot)\) \(\chi_{5520}(1021,\cdot)\) \(\chi_{5520}(1501,\cdot)\) \(\chi_{5520}(1741,\cdot)\) \(\chi_{5520}(1981,\cdot)\) \(\chi_{5520}(2101,\cdot)\) \(\chi_{5520}(2221,\cdot)\) \(\chi_{5520}(2341,\cdot)\) \(\chi_{5520}(3061,\cdot)\) \(\chi_{5520}(3301,\cdot)\) \(\chi_{5520}(3661,\cdot)\) \(\chi_{5520}(3781,\cdot)\) \(\chi_{5520}(4261,\cdot)\) \(\chi_{5520}(4501,\cdot)\) \(\chi_{5520}(4741,\cdot)\) \(\chi_{5520}(4861,\cdot)\) \(\chi_{5520}(4981,\cdot)\) \(\chi_{5520}(5101,\cdot)\)
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: | 44.44.7829660228065619245582194641412012312544945884150589900838471630076269829766255604192509952.1 |
Values on generators
\((4831,1381,1841,4417,1201)\) → \((1,i,1,1,e\left(\frac{9}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5520 }(4261, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{15}{44}\right)\) |
sage: chi.jacobi_sum(n)