from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6030, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,0,64]))
pari: [g,chi] = znchar(Mod(151,6030))
Basic properties
| Modulus: | \(6030\) | |
| Conductor: | \(603\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{603}(151,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6030.cs
\(\chi_{6030}(151,\cdot)\) \(\chi_{6030}(301,\cdot)\) \(\chi_{6030}(391,\cdot)\) \(\chi_{6030}(421,\cdot)\) \(\chi_{6030}(571,\cdot)\) \(\chi_{6030}(601,\cdot)\) \(\chi_{6030}(931,\cdot)\) \(\chi_{6030}(961,\cdot)\) \(\chi_{6030}(1681,\cdot)\) \(\chi_{6030}(2131,\cdot)\) \(\chi_{6030}(2191,\cdot)\) \(\chi_{6030}(2221,\cdot)\) \(\chi_{6030}(2371,\cdot)\) \(\chi_{6030}(3031,\cdot)\) \(\chi_{6030}(3121,\cdot)\) \(\chi_{6030}(3271,\cdot)\) \(\chi_{6030}(3721,\cdot)\) \(\chi_{6030}(4471,\cdot)\) \(\chi_{6030}(4711,\cdot)\) \(\chi_{6030}(5431,\cdot)\)
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 33 polynomial |
Values on generators
\((4691,1207,3151)\) → \((e\left(\frac{2}{3}\right),1,e\left(\frac{32}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6030 }(151, a) \) | \(1\) | \(1\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{11}\right)\) |
sage: chi.jacobi_sum(n)