from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9072, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,28,0]))
pari: [g,chi] = znchar(Mod(3277,9072))
Basic properties
Modulus: | \(9072\) | |
Conductor: | \(432\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{432}(157,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9072.gv
\(\chi_{9072}(253,\cdot)\) \(\chi_{9072}(1261,\cdot)\) \(\chi_{9072}(1765,\cdot)\) \(\chi_{9072}(2773,\cdot)\) \(\chi_{9072}(3277,\cdot)\) \(\chi_{9072}(4285,\cdot)\) \(\chi_{9072}(4789,\cdot)\) \(\chi_{9072}(5797,\cdot)\) \(\chi_{9072}(6301,\cdot)\) \(\chi_{9072}(7309,\cdot)\) \(\chi_{9072}(7813,\cdot)\) \(\chi_{9072}(8821,\cdot)\)
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: | 36.36.614667125325361522818798575155151578949632894783197825857500612833312768.1 |
Values on generators
\((1135,6805,3809,2593)\) → \((1,-i,e\left(\frac{7}{9}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 9072 }(3277, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)