from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,44,27]))
pari: [g,chi] = znchar(Mod(11,2415))
Basic properties
| Modulus: | \(2415\) | |
| Conductor: | \(483\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{483}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2415.da
\(\chi_{2415}(11,\cdot)\) \(\chi_{2415}(86,\cdot)\) \(\chi_{2415}(191,\cdot)\) \(\chi_{2415}(221,\cdot)\) \(\chi_{2415}(296,\cdot)\) \(\chi_{2415}(401,\cdot)\) \(\chi_{2415}(431,\cdot)\) \(\chi_{2415}(536,\cdot)\) \(\chi_{2415}(641,\cdot)\) \(\chi_{2415}(746,\cdot)\) \(\chi_{2415}(1031,\cdot)\) \(\chi_{2415}(1376,\cdot)\) \(\chi_{2415}(1556,\cdot)\) \(\chi_{2415}(1661,\cdot)\) \(\chi_{2415}(1901,\cdot)\) \(\chi_{2415}(1976,\cdot)\) \(\chi_{2415}(2006,\cdot)\) \(\chi_{2415}(2081,\cdot)\) \(\chi_{2415}(2291,\cdot)\) \(\chi_{2415}(2321,\cdot)\)
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
\((806,967,346,1891)\) → \((-1,1,e\left(\frac{2}{3}\right),e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(11, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(-1\) | \(e\left(\frac{25}{66}\right)\) |
sage: chi.jacobi_sum(n)