sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(1323, base_ring=CyclotomicField(42))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([14,30]))
pari: [g,chi] = znchar(Mod(64,1323))
Basic properties
| Modulus: | \(1323\) | |
| Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{441}(211,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1323.bm
\(\chi_{1323}(64,\cdot)\) \(\chi_{1323}(127,\cdot)\) \(\chi_{1323}(253,\cdot)\) \(\chi_{1323}(316,\cdot)\) \(\chi_{1323}(505,\cdot)\) \(\chi_{1323}(631,\cdot)\) \(\chi_{1323}(694,\cdot)\) \(\chi_{1323}(820,\cdot)\) \(\chi_{1323}(1009,\cdot)\) \(\chi_{1323}(1072,\cdot)\) \(\chi_{1323}(1198,\cdot)\) \(\chi_{1323}(1261,\cdot)\)
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
\((785,1081)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{5}{7}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \(1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(1\) |
Related number fields
| Field of values: | \(\Q(\zeta_{21})\) |
| Fixed field: | 21.21.60663096207149471029120391038078960708572561.1 |