from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9072, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,27,1,0]))
pari: [g,chi] = znchar(Mod(407,9072))
Basic properties
| Modulus: | \(9072\) | |
| Conductor: | \(648\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{648}(83,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9072.ii
\(\chi_{9072}(407,\cdot)\) \(\chi_{9072}(743,\cdot)\) \(\chi_{9072}(1415,\cdot)\) \(\chi_{9072}(1751,\cdot)\) \(\chi_{9072}(2423,\cdot)\) \(\chi_{9072}(2759,\cdot)\) \(\chi_{9072}(3431,\cdot)\) \(\chi_{9072}(3767,\cdot)\) \(\chi_{9072}(4439,\cdot)\) \(\chi_{9072}(4775,\cdot)\) \(\chi_{9072}(5447,\cdot)\) \(\chi_{9072}(5783,\cdot)\) \(\chi_{9072}(6455,\cdot)\) \(\chi_{9072}(6791,\cdot)\) \(\chi_{9072}(7463,\cdot)\) \(\chi_{9072}(7799,\cdot)\) \(\chi_{9072}(8471,\cdot)\) \(\chi_{9072}(8807,\cdot)\)
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
\((1135,6805,3809,2593)\) → \((-1,-1,e\left(\frac{1}{54}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 9072 }(407, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{5}{18}\right)\) |
sage: chi.jacobi_sum(n)