from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,55,45]))
pari: [g,chi] = znchar(Mod(19,2415))
Basic properties
Modulus: | \(2415\) | |
Conductor: | \(805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{805}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2415.cv
\(\chi_{2415}(19,\cdot)\) \(\chi_{2415}(199,\cdot)\) \(\chi_{2415}(304,\cdot)\) \(\chi_{2415}(544,\cdot)\) \(\chi_{2415}(619,\cdot)\) \(\chi_{2415}(649,\cdot)\) \(\chi_{2415}(724,\cdot)\) \(\chi_{2415}(934,\cdot)\) \(\chi_{2415}(964,\cdot)\) \(\chi_{2415}(1069,\cdot)\) \(\chi_{2415}(1144,\cdot)\) \(\chi_{2415}(1249,\cdot)\) \(\chi_{2415}(1279,\cdot)\) \(\chi_{2415}(1354,\cdot)\) \(\chi_{2415}(1459,\cdot)\) \(\chi_{2415}(1489,\cdot)\) \(\chi_{2415}(1594,\cdot)\) \(\chi_{2415}(1699,\cdot)\) \(\chi_{2415}(1804,\cdot)\) \(\chi_{2415}(2089,\cdot)\)
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
\((806,967,346,1891)\) → \((1,-1,e\left(\frac{5}{6}\right),e\left(\frac{15}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
\( \chi_{ 2415 }(19, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(1\) | \(e\left(\frac{5}{66}\right)\) |
sage: chi.jacobi_sum(n)