from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,18,0]))
pari: [g,chi] = znchar(Mod(3481,6003))
Basic properties
Modulus: | \(6003\) | |
Conductor: | \(207\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{207}(169,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6003.bt
\(\chi_{6003}(349,\cdot)\) \(\chi_{6003}(610,\cdot)\) \(\chi_{6003}(1393,\cdot)\) \(\chi_{6003}(1480,\cdot)\) \(\chi_{6003}(1741,\cdot)\) \(\chi_{6003}(1915,\cdot)\) \(\chi_{6003}(2263,\cdot)\) \(\chi_{6003}(2785,\cdot)\) \(\chi_{6003}(3307,\cdot)\) \(\chi_{6003}(3481,\cdot)\) \(\chi_{6003}(3568,\cdot)\) \(\chi_{6003}(3742,\cdot)\) \(\chi_{6003}(4264,\cdot)\) \(\chi_{6003}(4351,\cdot)\) \(\chi_{6003}(4612,\cdot)\) \(\chi_{6003}(4786,\cdot)\) \(\chi_{6003}(5308,\cdot)\) \(\chi_{6003}(5395,\cdot)\) \(\chi_{6003}(5569,\cdot)\) \(\chi_{6003}(5917,\cdot)\)
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: | 33.33.70011645999218458416472683122408534303895571350166174758601569.1 |
Values on generators
\((668,3133,4555)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{3}{11}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 6003 }(3481, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) |
sage: chi.jacobi_sum(n)