from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9072, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,22,30]))
pari: [g,chi] = znchar(Mod(341,9072))
Basic properties
| Modulus: | \(9072\) | |
| Conductor: | \(3024\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3024}(2021,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9072.hl
\(\chi_{9072}(341,\cdot)\) \(\chi_{9072}(1277,\cdot)\) \(\chi_{9072}(1853,\cdot)\) \(\chi_{9072}(2789,\cdot)\) \(\chi_{9072}(3365,\cdot)\) \(\chi_{9072}(4301,\cdot)\) \(\chi_{9072}(4877,\cdot)\) \(\chi_{9072}(5813,\cdot)\) \(\chi_{9072}(6389,\cdot)\) \(\chi_{9072}(7325,\cdot)\) \(\chi_{9072}(7901,\cdot)\) \(\chi_{9072}(8837,\cdot)\)
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: | 36.36.124687723528889177570420723064261381086672513377851660184757530334936556972919209118584893066969088.2 |
Values on generators
\((1135,6805,3809,2593)\) → \((1,i,e\left(\frac{11}{18}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 9072 }(341, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(1\) | \(i\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)