from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,52]))
pari: [g,chi] = znchar(Mod(61,324))
Basic properties
| Modulus: | \(324\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(27\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{81}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 324.m
\(\chi_{324}(13,\cdot)\) \(\chi_{324}(25,\cdot)\) \(\chi_{324}(49,\cdot)\) \(\chi_{324}(61,\cdot)\) \(\chi_{324}(85,\cdot)\) \(\chi_{324}(97,\cdot)\) \(\chi_{324}(121,\cdot)\) \(\chi_{324}(133,\cdot)\) \(\chi_{324}(157,\cdot)\) \(\chi_{324}(169,\cdot)\) \(\chi_{324}(193,\cdot)\) \(\chi_{324}(205,\cdot)\) \(\chi_{324}(229,\cdot)\) \(\chi_{324}(241,\cdot)\) \(\chi_{324}(265,\cdot)\) \(\chi_{324}(277,\cdot)\) \(\chi_{324}(301,\cdot)\) \(\chi_{324}(313,\cdot)\)
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 27 polynomial |
Values on generators
\((163,245)\) → \((1,e\left(\frac{26}{27}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 324 }(61, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) |
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)