from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5445, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,0,45]))
pari: [g,chi] = znchar(Mod(131,5445))
Basic properties
| Modulus: | \(5445\) | |
| Conductor: | \(1089\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1089}(131,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5445.cp
\(\chi_{5445}(131,\cdot)\) \(\chi_{5445}(461,\cdot)\) \(\chi_{5445}(626,\cdot)\) \(\chi_{5445}(956,\cdot)\) \(\chi_{5445}(1121,\cdot)\) \(\chi_{5445}(1616,\cdot)\) \(\chi_{5445}(1946,\cdot)\) \(\chi_{5445}(2111,\cdot)\) \(\chi_{5445}(2441,\cdot)\) \(\chi_{5445}(2606,\cdot)\) \(\chi_{5445}(2936,\cdot)\) \(\chi_{5445}(3101,\cdot)\) \(\chi_{5445}(3431,\cdot)\) \(\chi_{5445}(3596,\cdot)\) \(\chi_{5445}(3926,\cdot)\) \(\chi_{5445}(4091,\cdot)\) \(\chi_{5445}(4421,\cdot)\) \(\chi_{5445}(4586,\cdot)\) \(\chi_{5445}(4916,\cdot)\) \(\chi_{5445}(5411,\cdot)\)
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
\((3026,4357,3511)\) → \((e\left(\frac{5}{6}\right),1,e\left(\frac{15}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 5445 }(131, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{59}{66}\right)\) |
sage: chi.jacobi_sum(n)