from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4560, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,9,14]))
pari: [g,chi] = znchar(Mod(337,4560))
Basic properties
Modulus: | \(4560\) | |
Conductor: | \(95\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{95}(52,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4560.hx
\(\chi_{4560}(97,\cdot)\) \(\chi_{4560}(193,\cdot)\) \(\chi_{4560}(337,\cdot)\) \(\chi_{4560}(433,\cdot)\) \(\chi_{4560}(1153,\cdot)\) \(\chi_{4560}(1777,\cdot)\) \(\chi_{4560}(2017,\cdot)\) \(\chi_{4560}(2257,\cdot)\) \(\chi_{4560}(2833,\cdot)\) \(\chi_{4560}(2977,\cdot)\) \(\chi_{4560}(3073,\cdot)\) \(\chi_{4560}(4513,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | \(\Q(\zeta_{95})^+\) |
Values on generators
\((1711,1141,3041,2737,1921)\) → \((1,1,1,i,e\left(\frac{7}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 4560 }(337, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-i\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) |
sage: chi.jacobi_sum(n)