from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8001, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,35,41]))
pari: [g,chi] = znchar(Mod(334,8001))
Basic properties
| Modulus: | \(8001\) | |
| 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 \(\chi_{889}(334,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8001.kl
\(\chi_{8001}(334,\cdot)\) \(\chi_{8001}(1153,\cdot)\) \(\chi_{8001}(1783,\cdot)\) \(\chi_{8001}(2098,\cdot)\) \(\chi_{8001}(2467,\cdot)\) \(\chi_{8001}(2845,\cdot)\) \(\chi_{8001}(4420,\cdot)\) \(\chi_{8001}(5374,\cdot)\) \(\chi_{8001}(5869,\cdot)\) \(\chi_{8001}(6058,\cdot)\) \(\chi_{8001}(6256,\cdot)\) \(\chi_{8001}(7570,\cdot)\)
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: | 42.42.68294221643735165246072560329301656207364612371718899040604677564104303244539355230507813039348036672652649670437161.1 |
Values on generators
\((3557,1144,7750)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{41}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 8001 }(334, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)