from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8800, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,15,16,10]))
pari: [g,chi] = znchar(Mod(4311,8800))
Basic properties
| Modulus: | \(8800\) | |
| Conductor: | \(4400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4400}(1011,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8800.gw
\(\chi_{8800}(791,\cdot)\) \(\chi_{8800}(1671,\cdot)\) \(\chi_{8800}(3431,\cdot)\) \(\chi_{8800}(4311,\cdot)\) \(\chi_{8800}(5191,\cdot)\) \(\chi_{8800}(6071,\cdot)\) \(\chi_{8800}(7831,\cdot)\) \(\chi_{8800}(8711,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((2751,3301,4577,5601)\) → \((-1,-i,e\left(\frac{4}{5}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8800 }(4311, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(-1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) |
sage: chi.jacobi_sum(n)