sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(1815, base_ring=CyclotomicField(22))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,11,10]))
pari: [g,chi] = znchar(Mod(34,1815))
Basic properties
| Modulus: | \(1815\) | |
| Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{605}(34,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1815.be
\(\chi_{1815}(34,\cdot)\) \(\chi_{1815}(199,\cdot)\) \(\chi_{1815}(529,\cdot)\) \(\chi_{1815}(694,\cdot)\) \(\chi_{1815}(859,\cdot)\) \(\chi_{1815}(1024,\cdot)\) \(\chi_{1815}(1189,\cdot)\) \(\chi_{1815}(1354,\cdot)\) \(\chi_{1815}(1519,\cdot)\) \(\chi_{1815}(1684,\cdot)\)
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
\((1211,727,1696)\) → \((1,-1,e\left(\frac{5}{11}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \(1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{11})\) |
| Fixed field: | 22.22.22099245882898413967719412126414511946210986328125.1 |