from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4025, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,11,48]))
pari: [g,chi] = znchar(Mod(2901,4025))
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 \(\chi_{161}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4025.cp
\(\chi_{4025}(26,\cdot)\) \(\chi_{4025}(101,\cdot)\) \(\chi_{4025}(376,\cdot)\) \(\chi_{4025}(726,\cdot)\) \(\chi_{4025}(901,\cdot)\) \(\chi_{4025}(1076,\cdot)\) \(\chi_{4025}(1251,\cdot)\) \(\chi_{4025}(1501,\cdot)\) \(\chi_{4025}(2026,\cdot)\) \(\chi_{4025}(2201,\cdot)\) \(\chi_{4025}(2651,\cdot)\) \(\chi_{4025}(2726,\cdot)\) \(\chi_{4025}(2901,\cdot)\) \(\chi_{4025}(3176,\cdot)\) \(\chi_{4025}(3251,\cdot)\) \(\chi_{4025}(3351,\cdot)\) \(\chi_{4025}(3601,\cdot)\) \(\chi_{4025}(3776,\cdot)\) \(\chi_{4025}(3876,\cdot)\) \(\chi_{4025}(3951,\cdot)\)
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
\((2577,1151,3501)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{8}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
\( \chi_{ 4025 }(2901, a) \) | \(-1\) | \(1\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{5}{33}\right)\) |
sage: chi.jacobi_sum(n)