from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8820, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,28,0,11]))
pari: [g,chi] = znchar(Mod(61,8820))
Basic properties
| Modulus: | \(8820\) | |
| Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{441}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8820.ft
\(\chi_{8820}(61,\cdot)\) \(\chi_{8820}(661,\cdot)\) \(\chi_{8820}(1321,\cdot)\) \(\chi_{8820}(1921,\cdot)\) \(\chi_{8820}(2581,\cdot)\) \(\chi_{8820}(3181,\cdot)\) \(\chi_{8820}(5101,\cdot)\) \(\chi_{8820}(5701,\cdot)\) \(\chi_{8820}(6361,\cdot)\) \(\chi_{8820}(6961,\cdot)\) \(\chi_{8820}(7621,\cdot)\) \(\chi_{8820}(8221,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((4411,7841,7057,1081)\) → \((1,e\left(\frac{2}{3}\right),1,e\left(\frac{11}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8820 }(61, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) |
sage: chi.jacobi_sum(n)