from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6030, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,31]))
pari: [g,chi] = znchar(Mod(251,6030))
Basic properties
| Modulus: | \(6030\) | |
| Conductor: | \(201\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{201}(50,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6030.dj
\(\chi_{6030}(251,\cdot)\) \(\chi_{6030}(701,\cdot)\) \(\chi_{6030}(1151,\cdot)\) \(\chi_{6030}(1961,\cdot)\) \(\chi_{6030}(2051,\cdot)\) \(\chi_{6030}(2231,\cdot)\) \(\chi_{6030}(3311,\cdot)\) \(\chi_{6030}(3401,\cdot)\) \(\chi_{6030}(3491,\cdot)\) \(\chi_{6030}(3851,\cdot)\) \(\chi_{6030}(4031,\cdot)\) \(\chi_{6030}(4121,\cdot)\) \(\chi_{6030}(4211,\cdot)\) \(\chi_{6030}(4301,\cdot)\) \(\chi_{6030}(4751,\cdot)\) \(\chi_{6030}(5021,\cdot)\) \(\chi_{6030}(5471,\cdot)\) \(\chi_{6030}(5741,\cdot)\) \(\chi_{6030}(5831,\cdot)\) \(\chi_{6030}(6011,\cdot)\)
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
\((4691,1207,3151)\) → \((-1,1,e\left(\frac{31}{66}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6030 }(251, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{33}\right)\) |
sage: chi.jacobi_sum(n)