from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2070, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,33,27]))
pari: [g,chi] = znchar(Mod(149,2070))
Basic properties
| Modulus: | \(2070\) | |
| Conductor: | \(1035\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1035}(149,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2070.br
\(\chi_{2070}(149,\cdot)\) \(\chi_{2070}(329,\cdot)\) \(\chi_{2070}(389,\cdot)\) \(\chi_{2070}(419,\cdot)\) \(\chi_{2070}(479,\cdot)\) \(\chi_{2070}(569,\cdot)\) \(\chi_{2070}(659,\cdot)\) \(\chi_{2070}(779,\cdot)\) \(\chi_{2070}(839,\cdot)\) \(\chi_{2070}(1019,\cdot)\) \(\chi_{2070}(1049,\cdot)\) \(\chi_{2070}(1109,\cdot)\) \(\chi_{2070}(1229,\cdot)\) \(\chi_{2070}(1469,\cdot)\) \(\chi_{2070}(1739,\cdot)\) \(\chi_{2070}(1769,\cdot)\) \(\chi_{2070}(1859,\cdot)\) \(\chi_{2070}(1919,\cdot)\) \(\chi_{2070}(1949,\cdot)\) \(\chi_{2070}(2039,\cdot)\)
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
\((461,1657,1891)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2070 }(149, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) |
sage: chi.jacobi_sum(n)