from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3240, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,0,53,0]))
pari: [g,chi] = znchar(Mod(41,3240))
Basic properties
| Modulus: | \(3240\) | |
| 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 \(\chi_{81}(41,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3240.dg
\(\chi_{3240}(41,\cdot)\) \(\chi_{3240}(281,\cdot)\) \(\chi_{3240}(401,\cdot)\) \(\chi_{3240}(641,\cdot)\) \(\chi_{3240}(761,\cdot)\) \(\chi_{3240}(1001,\cdot)\) \(\chi_{3240}(1121,\cdot)\) \(\chi_{3240}(1361,\cdot)\) \(\chi_{3240}(1481,\cdot)\) \(\chi_{3240}(1721,\cdot)\) \(\chi_{3240}(1841,\cdot)\) \(\chi_{3240}(2081,\cdot)\) \(\chi_{3240}(2201,\cdot)\) \(\chi_{3240}(2441,\cdot)\) \(\chi_{3240}(2561,\cdot)\) \(\chi_{3240}(2801,\cdot)\) \(\chi_{3240}(2921,\cdot)\) \(\chi_{3240}(3161,\cdot)\)
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
\((2431,1621,3161,1297)\) → \((1,1,e\left(\frac{53}{54}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3240 }(41, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{54}\right)\) |
sage: chi.jacobi_sum(n)