from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,55,48]))
pari: [g,chi] = znchar(Mod(26,2415))
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 \(\chi_{483}(26,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2415.cy
\(\chi_{2415}(26,\cdot)\) \(\chi_{2415}(101,\cdot)\) \(\chi_{2415}(131,\cdot)\) \(\chi_{2415}(236,\cdot)\) \(\chi_{2415}(311,\cdot)\) \(\chi_{2415}(416,\cdot)\) \(\chi_{2415}(446,\cdot)\) \(\chi_{2415}(656,\cdot)\) \(\chi_{2415}(731,\cdot)\) \(\chi_{2415}(761,\cdot)\) \(\chi_{2415}(836,\cdot)\) \(\chi_{2415}(1076,\cdot)\) \(\chi_{2415}(1181,\cdot)\) \(\chi_{2415}(1361,\cdot)\) \(\chi_{2415}(1706,\cdot)\) \(\chi_{2415}(1991,\cdot)\) \(\chi_{2415}(2096,\cdot)\) \(\chi_{2415}(2201,\cdot)\) \(\chi_{2415}(2306,\cdot)\) \(\chi_{2415}(2336,\cdot)\)
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
\((806,967,346,1891)\) → \((-1,1,e\left(\frac{5}{6}\right),e\left(\frac{8}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(26, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(1\) | \(e\left(\frac{10}{33}\right)\) |
sage: chi.jacobi_sum(n)