from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,51,25]))
pari: [g,chi] = znchar(Mod(97,7800))
Basic properties
| Modulus: | \(7800\) | |
| Conductor: | \(325\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{325}(97,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7800.mb
\(\chi_{7800}(97,\cdot)\) \(\chi_{7800}(553,\cdot)\) \(\chi_{7800}(1753,\cdot)\) \(\chi_{7800}(2017,\cdot)\) \(\chi_{7800}(2113,\cdot)\) \(\chi_{7800}(3217,\cdot)\) \(\chi_{7800}(3313,\cdot)\) \(\chi_{7800}(3577,\cdot)\) \(\chi_{7800}(3673,\cdot)\) \(\chi_{7800}(4777,\cdot)\) \(\chi_{7800}(4873,\cdot)\) \(\chi_{7800}(5137,\cdot)\) \(\chi_{7800}(5233,\cdot)\) \(\chi_{7800}(6337,\cdot)\) \(\chi_{7800}(6433,\cdot)\) \(\chi_{7800}(6697,\cdot)\)
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
\((1951,3901,5201,7177,4201)\) → \((1,1,1,e\left(\frac{17}{20}\right),e\left(\frac{5}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7800 }(97, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)