from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(670, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,62]))
pari: [g,chi] = znchar(Mod(21,670))
Basic properties
| Modulus: | \(670\) | |
| Conductor: | \(67\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{67}(21,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 670.q
\(\chi_{670}(21,\cdot)\) \(\chi_{670}(71,\cdot)\) \(\chi_{670}(121,\cdot)\) \(\chi_{670}(151,\cdot)\) \(\chi_{670}(181,\cdot)\) \(\chi_{670}(211,\cdot)\) \(\chi_{670}(261,\cdot)\) \(\chi_{670}(291,\cdot)\) \(\chi_{670}(301,\cdot)\) \(\chi_{670}(341,\cdot)\) \(\chi_{670}(351,\cdot)\) \(\chi_{670}(361,\cdot)\) \(\chi_{670}(371,\cdot)\) \(\chi_{670}(391,\cdot)\) \(\chi_{670}(421,\cdot)\) \(\chi_{670}(441,\cdot)\) \(\chi_{670}(451,\cdot)\) \(\chi_{670}(571,\cdot)\) \(\chi_{670}(591,\cdot)\) \(\chi_{670}(601,\cdot)\)
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 33 polynomial |
Values on generators
\((537,471)\) → \((1,e\left(\frac{31}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 670 }(21, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) |
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)