from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1380, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,22,11,38]))
pari: [g,chi] = znchar(Mod(467,1380))
Basic properties
Modulus: | \(1380\) | |
Conductor: | \(1380\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1380.bv
\(\chi_{1380}(83,\cdot)\) \(\chi_{1380}(107,\cdot)\) \(\chi_{1380}(143,\cdot)\) \(\chi_{1380}(203,\cdot)\) \(\chi_{1380}(227,\cdot)\) \(\chi_{1380}(263,\cdot)\) \(\chi_{1380}(287,\cdot)\) \(\chi_{1380}(383,\cdot)\) \(\chi_{1380}(467,\cdot)\) \(\chi_{1380}(503,\cdot)\) \(\chi_{1380}(527,\cdot)\) \(\chi_{1380}(563,\cdot)\) \(\chi_{1380}(707,\cdot)\) \(\chi_{1380}(743,\cdot)\) \(\chi_{1380}(803,\cdot)\) \(\chi_{1380}(983,\cdot)\) \(\chi_{1380}(1187,\cdot)\) \(\chi_{1380}(1247,\cdot)\) \(\chi_{1380}(1307,\cdot)\) \(\chi_{1380}(1367,\cdot)\)
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
\((691,461,277,1201)\) → \((-1,-1,i,e\left(\frac{19}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 1380 }(467, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{25}{44}\right)\) |
sage: chi.jacobi_sum(n)