sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(1323, base_ring=CyclotomicField(42))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([7,3]))
pari: [g,chi] = znchar(Mod(1007,1323))
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 \(\chi_{441}(272,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1323.bt
\(\chi_{1323}(62,\cdot)\) \(\chi_{1323}(125,\cdot)\) \(\chi_{1323}(251,\cdot)\) \(\chi_{1323}(314,\cdot)\) \(\chi_{1323}(503,\cdot)\) \(\chi_{1323}(629,\cdot)\) \(\chi_{1323}(692,\cdot)\) \(\chi_{1323}(818,\cdot)\) \(\chi_{1323}(1007,\cdot)\) \(\chi_{1323}(1070,\cdot)\) \(\chi_{1323}(1196,\cdot)\) \(\chi_{1323}(1259,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.42.2760527312663447143294115190995836828884368283864639835402535675082795095484429706956536669661.1 |
Values on generators
\((785,1081)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{14}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(-1\) |