from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3645, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([37,0]))
pari: [g,chi] = znchar(Mod(26,3645))
Basic properties
| Modulus: | \(3645\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{81}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3645.u
\(\chi_{3645}(26,\cdot)\) \(\chi_{3645}(296,\cdot)\) \(\chi_{3645}(431,\cdot)\) \(\chi_{3645}(701,\cdot)\) \(\chi_{3645}(836,\cdot)\) \(\chi_{3645}(1106,\cdot)\) \(\chi_{3645}(1241,\cdot)\) \(\chi_{3645}(1511,\cdot)\) \(\chi_{3645}(1646,\cdot)\) \(\chi_{3645}(1916,\cdot)\) \(\chi_{3645}(2051,\cdot)\) \(\chi_{3645}(2321,\cdot)\) \(\chi_{3645}(2456,\cdot)\) \(\chi_{3645}(2726,\cdot)\) \(\chi_{3645}(2861,\cdot)\) \(\chi_{3645}(3131,\cdot)\) \(\chi_{3645}(3266,\cdot)\) \(\chi_{3645}(3536,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((731,2917)\) → \((e\left(\frac{37}{54}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3645 }(26, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) |
sage: chi.jacobi_sum(n)