from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1380, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,0,11,26]))
pari: [g,chi] = znchar(Mod(757,1380))
Basic properties
Modulus: | \(1380\) | |
Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{115}(67,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1380.bs
\(\chi_{1380}(37,\cdot)\) \(\chi_{1380}(97,\cdot)\) \(\chi_{1380}(157,\cdot)\) \(\chi_{1380}(217,\cdot)\) \(\chi_{1380}(313,\cdot)\) \(\chi_{1380}(337,\cdot)\) \(\chi_{1380}(373,\cdot)\) \(\chi_{1380}(433,\cdot)\) \(\chi_{1380}(457,\cdot)\) \(\chi_{1380}(493,\cdot)\) \(\chi_{1380}(517,\cdot)\) \(\chi_{1380}(613,\cdot)\) \(\chi_{1380}(697,\cdot)\) \(\chi_{1380}(733,\cdot)\) \(\chi_{1380}(757,\cdot)\) \(\chi_{1380}(793,\cdot)\) \(\chi_{1380}(937,\cdot)\) \(\chi_{1380}(973,\cdot)\) \(\chi_{1380}(1033,\cdot)\) \(\chi_{1380}(1213,\cdot)\)
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: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((691,461,277,1201)\) → \((1,1,i,e\left(\frac{13}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 1380 }(757, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) |
sage: chi.jacobi_sum(n)