from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1900, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,32]))
pari: [g,chi] = znchar(Mod(1107,1900))
Basic properties
| Modulus: | \(1900\) | |
| Conductor: | \(380\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{380}(347,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1900.cd
\(\chi_{1900}(43,\cdot)\) \(\chi_{1900}(207,\cdot)\) \(\chi_{1900}(443,\cdot)\) \(\chi_{1900}(643,\cdot)\) \(\chi_{1900}(707,\cdot)\) \(\chi_{1900}(807,\cdot)\) \(\chi_{1900}(1043,\cdot)\) \(\chi_{1900}(1107,\cdot)\) \(\chi_{1900}(1507,\cdot)\) \(\chi_{1900}(1543,\cdot)\) \(\chi_{1900}(1643,\cdot)\) \(\chi_{1900}(1707,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((951,77,401)\) → \((-1,i,e\left(\frac{8}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1900 }(1107, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) |
sage: chi.jacobi_sum(n)