from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3240, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,18,26,9]))
pari: [g,chi] = znchar(Mod(197,3240))
Basic properties
| Modulus: | \(3240\) | |
| Conductor: | \(1080\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1080}(1037,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3240.cp
\(\chi_{3240}(197,\cdot)\) \(\chi_{3240}(413,\cdot)\) \(\chi_{3240}(557,\cdot)\) \(\chi_{3240}(773,\cdot)\) \(\chi_{3240}(1277,\cdot)\) \(\chi_{3240}(1493,\cdot)\) \(\chi_{3240}(1637,\cdot)\) \(\chi_{3240}(1853,\cdot)\) \(\chi_{3240}(2357,\cdot)\) \(\chi_{3240}(2573,\cdot)\) \(\chi_{3240}(2717,\cdot)\) \(\chi_{3240}(2933,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.36.1171447440175506066520511744440292972455846710607872000000000000000000000000000.1 |
Values on generators
\((2431,1621,3161,1297)\) → \((1,-1,e\left(\frac{13}{18}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3240 }(197, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) |
sage: chi.jacobi_sum(n)