from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3240, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,27,52,0]))
pari: [g,chi] = znchar(Mod(61,3240))
Basic properties
| Modulus: | \(3240\) | |
| Conductor: | \(648\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{648}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3240.dj
\(\chi_{3240}(61,\cdot)\) \(\chi_{3240}(301,\cdot)\) \(\chi_{3240}(421,\cdot)\) \(\chi_{3240}(661,\cdot)\) \(\chi_{3240}(781,\cdot)\) \(\chi_{3240}(1021,\cdot)\) \(\chi_{3240}(1141,\cdot)\) \(\chi_{3240}(1381,\cdot)\) \(\chi_{3240}(1501,\cdot)\) \(\chi_{3240}(1741,\cdot)\) \(\chi_{3240}(1861,\cdot)\) \(\chi_{3240}(2101,\cdot)\) \(\chi_{3240}(2221,\cdot)\) \(\chi_{3240}(2461,\cdot)\) \(\chi_{3240}(2581,\cdot)\) \(\chi_{3240}(2821,\cdot)\) \(\chi_{3240}(2941,\cdot)\) \(\chi_{3240}(3181,\cdot)\)
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
\((2431,1621,3161,1297)\) → \((1,-1,e\left(\frac{26}{27}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3240 }(61, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{27}\right)\) |
sage: chi.jacobi_sum(n)