from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4025, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,22,24]))
pari: [g,chi] = znchar(Mod(1168,4025))
Basic properties
| Modulus: | \(4025\) | |
| Conductor: | \(805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{805}(363,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4025.cd
\(\chi_{4025}(118,\cdot)\) \(\chi_{4025}(307,\cdot)\) \(\chi_{4025}(468,\cdot)\) \(\chi_{4025}(657,\cdot)\) \(\chi_{4025}(818,\cdot)\) \(\chi_{4025}(832,\cdot)\) \(\chi_{4025}(993,\cdot)\) \(\chi_{4025}(1007,\cdot)\) \(\chi_{4025}(1168,\cdot)\) \(\chi_{4025}(1182,\cdot)\) \(\chi_{4025}(1343,\cdot)\) \(\chi_{4025}(2582,\cdot)\) \(\chi_{4025}(2743,\cdot)\) \(\chi_{4025}(3107,\cdot)\) \(\chi_{4025}(3268,\cdot)\) \(\chi_{4025}(3282,\cdot)\) \(\chi_{4025}(3443,\cdot)\) \(\chi_{4025}(3807,\cdot)\) \(\chi_{4025}(3968,\cdot)\) \(\chi_{4025}(3982,\cdot)\)
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
\((2577,1151,3501)\) → \((-i,-1,e\left(\frac{6}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 4025 }(1168, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) |
sage: chi.jacobi_sum(n)