from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3025, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,8]))
pari: [g,chi] = znchar(Mod(232,3025))
Basic properties
| Modulus: | \(3025\) | |
| Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{605}(232,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3025.bu
\(\chi_{3025}(232,\cdot)\) \(\chi_{3025}(507,\cdot)\) \(\chi_{3025}(518,\cdot)\) \(\chi_{3025}(782,\cdot)\) \(\chi_{3025}(793,\cdot)\) \(\chi_{3025}(1057,\cdot)\) \(\chi_{3025}(1068,\cdot)\) \(\chi_{3025}(1343,\cdot)\) \(\chi_{3025}(1607,\cdot)\) \(\chi_{3025}(1618,\cdot)\) \(\chi_{3025}(1882,\cdot)\) \(\chi_{3025}(1893,\cdot)\) \(\chi_{3025}(2157,\cdot)\) \(\chi_{3025}(2168,\cdot)\) \(\chi_{3025}(2432,\cdot)\) \(\chi_{3025}(2443,\cdot)\) \(\chi_{3025}(2707,\cdot)\) \(\chi_{3025}(2718,\cdot)\) \(\chi_{3025}(2982,\cdot)\) \(\chi_{3025}(2993,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.0.23846517021132934583872858710431976074170086834253431126547411945070717739767846278962679207324981689453125.1 |
Values on generators
\((727,2301)\) → \((i,e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 3025 }(232, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{44}\right)\) | \(-i\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(-1\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) |
sage: chi.jacobi_sum(n)