from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4025, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,55,51]))
pari: [g,chi] = znchar(Mod(2476,4025))
Basic properties
| Modulus: | \(4025\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{161}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4025.co
\(\chi_{4025}(201,\cdot)\) \(\chi_{4025}(451,\cdot)\) \(\chi_{4025}(626,\cdot)\) \(\chi_{4025}(801,\cdot)\) \(\chi_{4025}(976,\cdot)\) \(\chi_{4025}(1326,\cdot)\) \(\chi_{4025}(1601,\cdot)\) \(\chi_{4025}(1676,\cdot)\) \(\chi_{4025}(1776,\cdot)\) \(\chi_{4025}(1851,\cdot)\) \(\chi_{4025}(1951,\cdot)\) \(\chi_{4025}(2126,\cdot)\) \(\chi_{4025}(2376,\cdot)\) \(\chi_{4025}(2476,\cdot)\) \(\chi_{4025}(2551,\cdot)\) \(\chi_{4025}(2826,\cdot)\) \(\chi_{4025}(3001,\cdot)\) \(\chi_{4025}(3076,\cdot)\) \(\chi_{4025}(3526,\cdot)\) \(\chi_{4025}(3701,\cdot)\)
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
\((2577,1151,3501)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{17}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 4025 }(2476, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{28}{33}\right)\) |
sage: chi.jacobi_sum(n)