from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,30,57,40]))
pari: [g,chi] = znchar(Mod(113,7800))
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 \(\chi_{975}(113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7800.kw
\(\chi_{7800}(113,\cdot)\) \(\chi_{7800}(497,\cdot)\) \(\chi_{7800}(737,\cdot)\) \(\chi_{7800}(1433,\cdot)\) \(\chi_{7800}(1673,\cdot)\) \(\chi_{7800}(2297,\cdot)\) \(\chi_{7800}(3233,\cdot)\) \(\chi_{7800}(3617,\cdot)\) \(\chi_{7800}(4553,\cdot)\) \(\chi_{7800}(5177,\cdot)\) \(\chi_{7800}(5417,\cdot)\) \(\chi_{7800}(6113,\cdot)\) \(\chi_{7800}(6353,\cdot)\) \(\chi_{7800}(6737,\cdot)\) \(\chi_{7800}(6977,\cdot)\) \(\chi_{7800}(7673,\cdot)\)
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
\((1951,3901,5201,7177,4201)\) → \((1,1,-1,e\left(\frac{19}{20}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7800 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)