from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3240, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,27,41,27]))
pari: [g,chi] = znchar(Mod(59,3240))
Basic properties
Modulus: | \(3240\) | |
Conductor: | \(3240\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3240.de
\(\chi_{3240}(59,\cdot)\) \(\chi_{3240}(299,\cdot)\) \(\chi_{3240}(419,\cdot)\) \(\chi_{3240}(659,\cdot)\) \(\chi_{3240}(779,\cdot)\) \(\chi_{3240}(1019,\cdot)\) \(\chi_{3240}(1139,\cdot)\) \(\chi_{3240}(1379,\cdot)\) \(\chi_{3240}(1499,\cdot)\) \(\chi_{3240}(1739,\cdot)\) \(\chi_{3240}(1859,\cdot)\) \(\chi_{3240}(2099,\cdot)\) \(\chi_{3240}(2219,\cdot)\) \(\chi_{3240}(2459,\cdot)\) \(\chi_{3240}(2579,\cdot)\) \(\chi_{3240}(2819,\cdot)\) \(\chi_{3240}(2939,\cdot)\) \(\chi_{3240}(3179,\cdot)\)
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
\((2431,1621,3161,1297)\) → \((-1,-1,e\left(\frac{41}{54}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3240 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{13}{54}\right)\) |
sage: chi.jacobi_sum(n)