from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9240, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,45,40,54]))
pari: [g,chi] = znchar(Mod(193,9240))
Basic properties
| Modulus: | \(9240\) | |
| Conductor: | \(385\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{385}(193,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9240.ok
\(\chi_{9240}(193,\cdot)\) \(\chi_{9240}(457,\cdot)\) \(\chi_{9240}(1537,\cdot)\) \(\chi_{9240}(2713,\cdot)\) \(\chi_{9240}(2977,\cdot)\) \(\chi_{9240}(3313,\cdot)\) \(\chi_{9240}(4153,\cdot)\) \(\chi_{9240}(4897,\cdot)\) \(\chi_{9240}(5233,\cdot)\) \(\chi_{9240}(5497,\cdot)\) \(\chi_{9240}(5737,\cdot)\) \(\chi_{9240}(6673,\cdot)\) \(\chi_{9240}(8257,\cdot)\) \(\chi_{9240}(8593,\cdot)\) \(\chi_{9240}(8857,\cdot)\) \(\chi_{9240}(9193,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((2311,4621,6161,3697,5281,2521)\) → \((1,1,1,-i,e\left(\frac{2}{3}\right),e\left(\frac{9}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 9240 }(193, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-i\) | \(e\left(\frac{17}{60}\right)\) |
sage: chi.jacobi_sum(n)