sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(1110, base_ring=CyclotomicField(36))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([18,9,26]))
pari: [g,chi] = znchar(Mod(77,1110))
Basic properties
| Modulus: | \(1110\) | |
| Conductor: | \(555\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{555}(77,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1110.cg
\(\chi_{1110}(77,\cdot)\) \(\chi_{1110}(173,\cdot)\) \(\chi_{1110}(263,\cdot)\) \(\chi_{1110}(287,\cdot)\) \(\chi_{1110}(317,\cdot)\) \(\chi_{1110}(437,\cdot)\) \(\chi_{1110}(617,\cdot)\) \(\chi_{1110}(707,\cdot)\) \(\chi_{1110}(743,\cdot)\) \(\chi_{1110}(953,\cdot)\) \(\chi_{1110}(983,\cdot)\) \(\chi_{1110}(1103,\cdot)\)
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
\((371,667,631)\) → \((-1,i,e\left(\frac{13}{18}\right))\)
Values
| \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
| \(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{17}{18}\right)\) | \(i\) |
Related number fields
| Field of values: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.36.601494028758700476950679105553655286733005172971144323588689573109149932861328125.1 |