from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8640, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,0,34,9]))
pari: [g,chi] = znchar(Mod(257,8640))
Basic properties
| Modulus: | \(8640\) | |
| Conductor: | \(135\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{135}(122,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8640.gg
\(\chi_{8640}(257,\cdot)\) \(\chi_{8640}(833,\cdot)\) \(\chi_{8640}(1217,\cdot)\) \(\chi_{8640}(1793,\cdot)\) \(\chi_{8640}(3137,\cdot)\) \(\chi_{8640}(3713,\cdot)\) \(\chi_{8640}(4097,\cdot)\) \(\chi_{8640}(4673,\cdot)\) \(\chi_{8640}(6017,\cdot)\) \(\chi_{8640}(6593,\cdot)\) \(\chi_{8640}(6977,\cdot)\) \(\chi_{8640}(7553,\cdot)\)
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: | \(\Q(\zeta_{135})^+\) |
Values on generators
\((2431,3781,6401,3457)\) → \((1,1,e\left(\frac{17}{18}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8640 }(257, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) |
sage: chi.jacobi_sum(n)