from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,22,38]))
pari: [g,chi] = znchar(Mod(2813,4830))
Basic properties
Modulus: | \(4830\) | |
Conductor: | \(2415\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2415}(398,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4830.cp
\(\chi_{4830}(83,\cdot)\) \(\chi_{4830}(293,\cdot)\) \(\chi_{4830}(503,\cdot)\) \(\chi_{4830}(797,\cdot)\) \(\chi_{4830}(1217,\cdot)\) \(\chi_{4830}(1763,\cdot)\) \(\chi_{4830}(1847,\cdot)\) \(\chi_{4830}(2057,\cdot)\) \(\chi_{4830}(2183,\cdot)\) \(\chi_{4830}(2687,\cdot)\) \(\chi_{4830}(2813,\cdot)\) \(\chi_{4830}(3023,\cdot)\) \(\chi_{4830}(3317,\cdot)\) \(\chi_{4830}(3653,\cdot)\) \(\chi_{4830}(3737,\cdot)\) \(\chi_{4830}(3947,\cdot)\) \(\chi_{4830}(4157,\cdot)\) \(\chi_{4830}(4283,\cdot)\) \(\chi_{4830}(4367,\cdot)\) \(\chi_{4830}(4703,\cdot)\)
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
\((3221,967,2761,1891)\) → \((-1,-i,-1,e\left(\frac{19}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 4830 }(2813, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)