sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(1323, base_ring=CyclotomicField(42))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([35,25]))
pari: [g,chi] = znchar(Mod(17,1323))
Basic properties
| Modulus: | \(1323\) | |
| Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{441}(311,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1323.bp
\(\chi_{1323}(17,\cdot)\) \(\chi_{1323}(89,\cdot)\) \(\chi_{1323}(206,\cdot)\) \(\chi_{1323}(278,\cdot)\) \(\chi_{1323}(395,\cdot)\) \(\chi_{1323}(467,\cdot)\) \(\chi_{1323}(584,\cdot)\) \(\chi_{1323}(773,\cdot)\) \(\chi_{1323}(845,\cdot)\) \(\chi_{1323}(1034,\cdot)\) \(\chi_{1323}(1151,\cdot)\) \(\chi_{1323}(1223,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Values on generators
\((785,1081)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{25}{42}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.42.135265838320508910021411644358796004615334045909367351934724248079056959678737055640870296813389.2 |