from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9072, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,0,17,45]))
pari: [g,chi] = znchar(Mod(257,9072))
Basic properties
| Modulus: | \(9072\) | |
| Conductor: | \(567\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{567}(257,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9072.hs
\(\chi_{9072}(257,\cdot)\) \(\chi_{9072}(353,\cdot)\) \(\chi_{9072}(1265,\cdot)\) \(\chi_{9072}(1361,\cdot)\) \(\chi_{9072}(2273,\cdot)\) \(\chi_{9072}(2369,\cdot)\) \(\chi_{9072}(3281,\cdot)\) \(\chi_{9072}(3377,\cdot)\) \(\chi_{9072}(4289,\cdot)\) \(\chi_{9072}(4385,\cdot)\) \(\chi_{9072}(5297,\cdot)\) \(\chi_{9072}(5393,\cdot)\) \(\chi_{9072}(6305,\cdot)\) \(\chi_{9072}(6401,\cdot)\) \(\chi_{9072}(7313,\cdot)\) \(\chi_{9072}(7409,\cdot)\) \(\chi_{9072}(8321,\cdot)\) \(\chi_{9072}(8417,\cdot)\)
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
\((1135,6805,3809,2593)\) → \((1,1,e\left(\frac{17}{54}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 9072 }(257, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{8}{9}\right)\) |
sage: chi.jacobi_sum(n)