from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8640, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,10,27]))
pari: [g,chi] = znchar(Mod(113,8640))
Basic properties
Modulus: | \(8640\) | |
Conductor: | \(2160\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2160}(1733,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8640.fy
\(\chi_{8640}(113,\cdot)\) \(\chi_{8640}(977,\cdot)\) \(\chi_{8640}(1073,\cdot)\) \(\chi_{8640}(1937,\cdot)\) \(\chi_{8640}(2993,\cdot)\) \(\chi_{8640}(3857,\cdot)\) \(\chi_{8640}(3953,\cdot)\) \(\chi_{8640}(4817,\cdot)\) \(\chi_{8640}(5873,\cdot)\) \(\chi_{8640}(6737,\cdot)\) \(\chi_{8640}(6833,\cdot)\) \(\chi_{8640}(7697,\cdot)\)
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: | 36.36.41216642617644769738384985747906299013992369570201489573102485504000000000000000000000000000.1 |
Values on generators
\((2431,3781,6401,3457)\) → \((1,i,e\left(\frac{5}{18}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 8640 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) |
sage: chi.jacobi_sum(n)