from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,22,14]))
pari: [g,chi] = znchar(Mod(1077,4235))
Basic properties
| Modulus: | \(4235\) | |
| Conductor: | \(4235\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4235.cc
\(\chi_{4235}(153,\cdot)\) \(\chi_{4235}(307,\cdot)\) \(\chi_{4235}(538,\cdot)\) \(\chi_{4235}(692,\cdot)\) \(\chi_{4235}(923,\cdot)\) \(\chi_{4235}(1077,\cdot)\) \(\chi_{4235}(1308,\cdot)\) \(\chi_{4235}(1462,\cdot)\) \(\chi_{4235}(1847,\cdot)\) \(\chi_{4235}(2078,\cdot)\) \(\chi_{4235}(2232,\cdot)\) \(\chi_{4235}(2463,\cdot)\) \(\chi_{4235}(2617,\cdot)\) \(\chi_{4235}(2848,\cdot)\) \(\chi_{4235}(3002,\cdot)\) \(\chi_{4235}(3233,\cdot)\) \(\chi_{4235}(3618,\cdot)\) \(\chi_{4235}(3772,\cdot)\) \(\chi_{4235}(4003,\cdot)\) \(\chi_{4235}(4157,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((2542,1816,2906)\) → \((i,-1,e\left(\frac{7}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 4235 }(1077, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(i\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(-1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) |
sage: chi.jacobi_sum(n)