from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5929, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,21]))
pari: [g,chi] = znchar(Mod(472,5929))
Basic properties
| Modulus: | \(5929\) | |
| Conductor: | \(847\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{847}(472,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5929.bg
\(\chi_{5929}(472,\cdot)\) \(\chi_{5929}(901,\cdot)\) \(\chi_{5929}(1011,\cdot)\) \(\chi_{5929}(1440,\cdot)\) \(\chi_{5929}(1550,\cdot)\) \(\chi_{5929}(1979,\cdot)\) \(\chi_{5929}(2089,\cdot)\) \(\chi_{5929}(2518,\cdot)\) \(\chi_{5929}(2628,\cdot)\) \(\chi_{5929}(3057,\cdot)\) \(\chi_{5929}(3167,\cdot)\) \(\chi_{5929}(3596,\cdot)\) \(\chi_{5929}(3706,\cdot)\) \(\chi_{5929}(4135,\cdot)\) \(\chi_{5929}(4245,\cdot)\) \(\chi_{5929}(4674,\cdot)\) \(\chi_{5929}(4784,\cdot)\) \(\chi_{5929}(5213,\cdot)\) \(\chi_{5929}(5752,\cdot)\) \(\chi_{5929}(5862,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((1816,2059)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{7}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 5929 }(472, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) |
sage: chi.jacobi_sum(n)