from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3240, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,0,32,27]))
pari: [g,chi] = znchar(Mod(49,3240))
Basic properties
Modulus: | \(3240\) | |
Conductor: | \(405\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{405}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3240.dh
\(\chi_{3240}(49,\cdot)\) \(\chi_{3240}(169,\cdot)\) \(\chi_{3240}(409,\cdot)\) \(\chi_{3240}(529,\cdot)\) \(\chi_{3240}(769,\cdot)\) \(\chi_{3240}(889,\cdot)\) \(\chi_{3240}(1129,\cdot)\) \(\chi_{3240}(1249,\cdot)\) \(\chi_{3240}(1489,\cdot)\) \(\chi_{3240}(1609,\cdot)\) \(\chi_{3240}(1849,\cdot)\) \(\chi_{3240}(1969,\cdot)\) \(\chi_{3240}(2209,\cdot)\) \(\chi_{3240}(2329,\cdot)\) \(\chi_{3240}(2569,\cdot)\) \(\chi_{3240}(2689,\cdot)\) \(\chi_{3240}(2929,\cdot)\) \(\chi_{3240}(3049,\cdot)\)
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
\((2431,1621,3161,1297)\) → \((1,1,e\left(\frac{16}{27}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3240 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{27}\right)\) |
sage: chi.jacobi_sum(n)