from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9800, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,15,21,25]))
pari: [g,chi] = znchar(Mod(7859,9800))
Basic properties
| Modulus: | \(9800\) | |
| Conductor: | \(1400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1400}(859,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9800.eh
\(\chi_{9800}(19,\cdot)\) \(\chi_{9800}(619,\cdot)\) \(\chi_{9800}(1979,\cdot)\) \(\chi_{9800}(2579,\cdot)\) \(\chi_{9800}(3939,\cdot)\) \(\chi_{9800}(4539,\cdot)\) \(\chi_{9800}(7859,\cdot)\) \(\chi_{9800}(8459,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.30.20954197182249451575109375000000000000000000000000000000000000000000000.1 |
Values on generators
\((7351,4901,1177,5001)\) → \((-1,-1,e\left(\frac{7}{10}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 9800 }(7859, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) |
sage: chi.jacobi_sum(n)