from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(74, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([31]))
pari: [g,chi] = znchar(Mod(59,74))
Basic properties
| Modulus: | \(74\) | |
| Conductor: | \(37\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{37}(22,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 74.i
\(\chi_{74}(5,\cdot)\) \(\chi_{74}(13,\cdot)\) \(\chi_{74}(15,\cdot)\) \(\chi_{74}(17,\cdot)\) \(\chi_{74}(19,\cdot)\) \(\chi_{74}(35,\cdot)\) \(\chi_{74}(39,\cdot)\) \(\chi_{74}(55,\cdot)\) \(\chi_{74}(57,\cdot)\) \(\chi_{74}(59,\cdot)\) \(\chi_{74}(61,\cdot)\) \(\chi_{74}(69,\cdot)\)
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
\(39\) → \(e\left(\frac{31}{36}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 74 }(59, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)