from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2070, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,0,18]))
pari: [g,chi] = znchar(Mod(31,2070))
Basic properties
| Modulus: | \(2070\) | |
| 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 \(\chi_{207}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2070.bg
\(\chi_{2070}(31,\cdot)\) \(\chi_{2070}(121,\cdot)\) \(\chi_{2070}(151,\cdot)\) \(\chi_{2070}(211,\cdot)\) \(\chi_{2070}(301,\cdot)\) \(\chi_{2070}(331,\cdot)\) \(\chi_{2070}(601,\cdot)\) \(\chi_{2070}(841,\cdot)\) \(\chi_{2070}(961,\cdot)\) \(\chi_{2070}(1021,\cdot)\) \(\chi_{2070}(1051,\cdot)\) \(\chi_{2070}(1231,\cdot)\) \(\chi_{2070}(1291,\cdot)\) \(\chi_{2070}(1411,\cdot)\) \(\chi_{2070}(1501,\cdot)\) \(\chi_{2070}(1591,\cdot)\) \(\chi_{2070}(1651,\cdot)\) \(\chi_{2070}(1681,\cdot)\) \(\chi_{2070}(1741,\cdot)\) \(\chi_{2070}(1921,\cdot)\)
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
\((461,1657,1891)\) → \((e\left(\frac{1}{3}\right),1,e\left(\frac{3}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2070 }(31, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) |
sage: chi.jacobi_sum(n)