from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,22,18]))
pari: [g,chi] = znchar(Mod(1066,2415))
Basic properties
Modulus: | \(2415\) | |
Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{161}(100,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2415.cm
\(\chi_{2415}(16,\cdot)\) \(\chi_{2415}(121,\cdot)\) \(\chi_{2415}(151,\cdot)\) \(\chi_{2415}(256,\cdot)\) \(\chi_{2415}(331,\cdot)\) \(\chi_{2415}(361,\cdot)\) \(\chi_{2415}(466,\cdot)\) \(\chi_{2415}(541,\cdot)\) \(\chi_{2415}(646,\cdot)\) \(\chi_{2415}(676,\cdot)\) \(\chi_{2415}(886,\cdot)\) \(\chi_{2415}(961,\cdot)\) \(\chi_{2415}(991,\cdot)\) \(\chi_{2415}(1066,\cdot)\) \(\chi_{2415}(1306,\cdot)\) \(\chi_{2415}(1411,\cdot)\) \(\chi_{2415}(1591,\cdot)\) \(\chi_{2415}(1936,\cdot)\) \(\chi_{2415}(2221,\cdot)\) \(\chi_{2415}(2326,\cdot)\)
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: | 33.33.277966181338944111003326058293667039541136678070715028736001.1 |
Values on generators
\((806,967,346,1891)\) → \((1,1,e\left(\frac{1}{3}\right),e\left(\frac{3}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
\( \chi_{ 2415 }(1066, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(1\) | \(e\left(\frac{1}{33}\right)\) |
sage: chi.jacobi_sum(n)