from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8001, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,7,39]))
pari: [g,chi] = znchar(Mod(1333,8001))
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 \(\chi_{889}(444,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8001.kn
\(\chi_{8001}(1333,\cdot)\) \(\chi_{8001}(1522,\cdot)\) \(\chi_{8001}(1774,\cdot)\) \(\chi_{8001}(2278,\cdot)\) \(\chi_{8001}(2476,\cdot)\) \(\chi_{8001}(2665,\cdot)\) \(\chi_{8001}(2917,\cdot)\) \(\chi_{8001}(3286,\cdot)\) \(\chi_{8001}(3421,\cdot)\) \(\chi_{8001}(4429,\cdot)\) \(\chi_{8001}(5302,\cdot)\) \(\chi_{8001}(6445,\cdot)\)
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: | 42.42.4234250210412001069258637257691218067292740552527676795871081751137969077099594223479931368302314878334220948009.1 |
Values on generators
\((3557,1144,7750)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{13}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 8001 }(1333, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)