from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3025, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,10]))
pari: [g,chi] = znchar(Mod(43,3025))
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 \(\chi_{605}(43,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3025.bv
\(\chi_{3025}(32,\cdot)\) \(\chi_{3025}(43,\cdot)\) \(\chi_{3025}(307,\cdot)\) \(\chi_{3025}(318,\cdot)\) \(\chi_{3025}(582,\cdot)\) \(\chi_{3025}(593,\cdot)\) \(\chi_{3025}(857,\cdot)\) \(\chi_{3025}(868,\cdot)\) \(\chi_{3025}(1132,\cdot)\) \(\chi_{3025}(1143,\cdot)\) \(\chi_{3025}(1407,\cdot)\) \(\chi_{3025}(1418,\cdot)\) \(\chi_{3025}(1682,\cdot)\) \(\chi_{3025}(1957,\cdot)\) \(\chi_{3025}(1968,\cdot)\) \(\chi_{3025}(2232,\cdot)\) \(\chi_{3025}(2243,\cdot)\) \(\chi_{3025}(2507,\cdot)\) \(\chi_{3025}(2518,\cdot)\) \(\chi_{3025}(2793,\cdot)\)
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: | 44.44.2885428559557085084648615903962269104974580506944665166312236845353556846511909399754484184086322784423828125.1 |
Values on generators
\((727,2301)\) → \((-i,e\left(\frac{5}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 3025 }(43, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{44}\right)\) | \(i\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(-1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) |
sage: chi.jacobi_sum(n)