from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9072, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,27,14,45]))
pari: [g,chi] = znchar(Mod(103,9072))
Basic properties
| Modulus: | \(9072\) | |
| Conductor: | \(4536\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4536}(2371,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9072.hp
\(\chi_{9072}(103,\cdot)\) \(\chi_{9072}(871,\cdot)\) \(\chi_{9072}(1111,\cdot)\) \(\chi_{9072}(1879,\cdot)\) \(\chi_{9072}(2119,\cdot)\) \(\chi_{9072}(2887,\cdot)\) \(\chi_{9072}(3127,\cdot)\) \(\chi_{9072}(3895,\cdot)\) \(\chi_{9072}(4135,\cdot)\) \(\chi_{9072}(4903,\cdot)\) \(\chi_{9072}(5143,\cdot)\) \(\chi_{9072}(5911,\cdot)\) \(\chi_{9072}(6151,\cdot)\) \(\chi_{9072}(6919,\cdot)\) \(\chi_{9072}(7159,\cdot)\) \(\chi_{9072}(7927,\cdot)\) \(\chi_{9072}(8167,\cdot)\) \(\chi_{9072}(8935,\cdot)\)
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
\((1135,6805,3809,2593)\) → \((-1,-1,e\left(\frac{7}{27}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 9072 }(103, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) |
sage: chi.jacobi_sum(n)