from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,30,18,35]))
pari: [g,chi] = znchar(Mod(89,7800))
Basic properties
| Modulus: | \(7800\) | |
| Conductor: | \(975\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{975}(89,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7800.lc
\(\chi_{7800}(89,\cdot)\) \(\chi_{7800}(929,\cdot)\) \(\chi_{7800}(1289,\cdot)\) \(\chi_{7800}(2009,\cdot)\) \(\chi_{7800}(2489,\cdot)\) \(\chi_{7800}(3209,\cdot)\) \(\chi_{7800}(3569,\cdot)\) \(\chi_{7800}(4409,\cdot)\) \(\chi_{7800}(4769,\cdot)\) \(\chi_{7800}(5129,\cdot)\) \(\chi_{7800}(5609,\cdot)\) \(\chi_{7800}(5969,\cdot)\) \(\chi_{7800}(6329,\cdot)\) \(\chi_{7800}(6689,\cdot)\) \(\chi_{7800}(7169,\cdot)\) \(\chi_{7800}(7529,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((1951,3901,5201,7177,4201)\) → \((1,1,-1,e\left(\frac{3}{10}\right),e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7800 }(89, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)