from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1449, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,33,36]))
pari: [g,chi] = znchar(Mod(41,1449))
Basic properties
| Modulus: | \(1449\) | |
| Conductor: | \(1449\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | 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 1449.cl
\(\chi_{1449}(41,\cdot)\) \(\chi_{1449}(104,\cdot)\) \(\chi_{1449}(146,\cdot)\) \(\chi_{1449}(167,\cdot)\) \(\chi_{1449}(209,\cdot)\) \(\chi_{1449}(335,\cdot)\) \(\chi_{1449}(524,\cdot)\) \(\chi_{1449}(545,\cdot)\) \(\chi_{1449}(587,\cdot)\) \(\chi_{1449}(650,\cdot)\) \(\chi_{1449}(671,\cdot)\) \(\chi_{1449}(860,\cdot)\) \(\chi_{1449}(923,\cdot)\) \(\chi_{1449}(1028,\cdot)\) \(\chi_{1449}(1112,\cdot)\) \(\chi_{1449}(1154,\cdot)\) \(\chi_{1449}(1175,\cdot)\) \(\chi_{1449}(1301,\cdot)\) \(\chi_{1449}(1343,\cdot)\) \(\chi_{1449}(1406,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((1289,829,442)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{6}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1449 }(41, a) \) | \(1\) | \(1\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)