from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3015, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,10]))
pari: [g,chi] = znchar(Mod(19,3015))
Basic properties
| Modulus: | \(3015\) | |
| Conductor: | \(335\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{335}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3015.df
\(\chi_{3015}(19,\cdot)\) \(\chi_{3015}(199,\cdot)\) \(\chi_{3015}(289,\cdot)\) \(\chi_{3015}(559,\cdot)\) \(\chi_{3015}(784,\cdot)\) \(\chi_{3015}(964,\cdot)\) \(\chi_{3015}(1009,\cdot)\) \(\chi_{3015}(1054,\cdot)\) \(\chi_{3015}(1279,\cdot)\) \(\chi_{3015}(1729,\cdot)\) \(\chi_{3015}(1819,\cdot)\) \(\chi_{3015}(1864,\cdot)\) \(\chi_{3015}(1909,\cdot)\) \(\chi_{3015}(1999,\cdot)\) \(\chi_{3015}(2179,\cdot)\) \(\chi_{3015}(2314,\cdot)\) \(\chi_{3015}(2539,\cdot)\) \(\chi_{3015}(2629,\cdot)\) \(\chi_{3015}(2719,\cdot)\) \(\chi_{3015}(2764,\cdot)\)
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
\((1676,1207,136)\) → \((1,-1,e\left(\frac{5}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3015 }(19, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) |
sage: chi.jacobi_sum(n)