from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,11,42]))
pari: [g,chi] = znchar(Mod(59,2415))
Basic properties
Modulus: | \(2415\) | |
Conductor: | \(2415\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2415.de
\(\chi_{2415}(59,\cdot)\) \(\chi_{2415}(164,\cdot)\) \(\chi_{2415}(269,\cdot)\) \(\chi_{2415}(374,\cdot)\) \(\chi_{2415}(404,\cdot)\) \(\chi_{2415}(509,\cdot)\) \(\chi_{2415}(584,\cdot)\) \(\chi_{2415}(614,\cdot)\) \(\chi_{2415}(719,\cdot)\) \(\chi_{2415}(794,\cdot)\) \(\chi_{2415}(899,\cdot)\) \(\chi_{2415}(929,\cdot)\) \(\chi_{2415}(1139,\cdot)\) \(\chi_{2415}(1214,\cdot)\) \(\chi_{2415}(1244,\cdot)\) \(\chi_{2415}(1319,\cdot)\) \(\chi_{2415}(1559,\cdot)\) \(\chi_{2415}(1664,\cdot)\) \(\chi_{2415}(1844,\cdot)\) \(\chi_{2415}(2189,\cdot)\)
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
\((806,967,346,1891)\) → \((-1,-1,e\left(\frac{1}{6}\right),e\left(\frac{7}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
\( \chi_{ 2415 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(-1\) | \(e\left(\frac{17}{33}\right)\) |
sage: chi.jacobi_sum(n)