from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,35]))
pari: [g,chi] = znchar(Mod(311,324))
Basic properties
| Modulus: | \(324\) | |
| Conductor: | \(324\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 324.p
\(\chi_{324}(11,\cdot)\) \(\chi_{324}(23,\cdot)\) \(\chi_{324}(47,\cdot)\) \(\chi_{324}(59,\cdot)\) \(\chi_{324}(83,\cdot)\) \(\chi_{324}(95,\cdot)\) \(\chi_{324}(119,\cdot)\) \(\chi_{324}(131,\cdot)\) \(\chi_{324}(155,\cdot)\) \(\chi_{324}(167,\cdot)\) \(\chi_{324}(191,\cdot)\) \(\chi_{324}(203,\cdot)\) \(\chi_{324}(227,\cdot)\) \(\chi_{324}(239,\cdot)\) \(\chi_{324}(263,\cdot)\) \(\chi_{324}(275,\cdot)\) \(\chi_{324}(299,\cdot)\) \(\chi_{324}(311,\cdot)\)
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
\((163,245)\) → \((-1,e\left(\frac{35}{54}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 324 }(311, a) \) | \(1\) | \(1\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{25}{54}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)