from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2667, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,7,11]))
pari: [g,chi] = znchar(Mod(10,2667))
Basic properties
| Modulus: | \(2667\) | |
| Conductor: | \(889\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{889}(10,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2667.du
\(\chi_{2667}(10,\cdot)\) \(\chi_{2667}(229,\cdot)\) \(\chi_{2667}(640,\cdot)\) \(\chi_{2667}(712,\cdot)\) \(\chi_{2667}(943,\cdot)\) \(\chi_{2667}(955,\cdot)\) \(\chi_{2667}(1321,\cdot)\) \(\chi_{2667}(1564,\cdot)\) \(\chi_{2667}(1678,\cdot)\) \(\chi_{2667}(1858,\cdot)\) \(\chi_{2667}(1867,\cdot)\) \(\chi_{2667}(2446,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.42.68294221643735165246072560329301656207364612371718899040604677564104303244539355230507813039348036672652649670437161.2 |
Values on generators
\((890,1144,2416)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{11}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 2667 }(10, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)