from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1380, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,11,26]))
pari: [g,chi] = znchar(Mod(1217,1380))
Basic properties
Modulus: | \(1380\) | |
Conductor: | \(345\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{345}(182,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1380.bo
\(\chi_{1380}(17,\cdot)\) \(\chi_{1380}(53,\cdot)\) \(\chi_{1380}(113,\cdot)\) \(\chi_{1380}(293,\cdot)\) \(\chi_{1380}(497,\cdot)\) \(\chi_{1380}(557,\cdot)\) \(\chi_{1380}(617,\cdot)\) \(\chi_{1380}(677,\cdot)\) \(\chi_{1380}(773,\cdot)\) \(\chi_{1380}(797,\cdot)\) \(\chi_{1380}(833,\cdot)\) \(\chi_{1380}(893,\cdot)\) \(\chi_{1380}(917,\cdot)\) \(\chi_{1380}(953,\cdot)\) \(\chi_{1380}(977,\cdot)\) \(\chi_{1380}(1073,\cdot)\) \(\chi_{1380}(1157,\cdot)\) \(\chi_{1380}(1193,\cdot)\) \(\chi_{1380}(1217,\cdot)\) \(\chi_{1380}(1253,\cdot)\)
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
\((691,461,277,1201)\) → \((1,-1,i,e\left(\frac{13}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 1380 }(1217, a) \) | \(-1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{31}{44}\right)\) |
sage: chi.jacobi_sum(n)