from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7581, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,19,26]))
pari: [g,chi] = znchar(Mod(20,7581))
Basic properties
| Modulus: | \(7581\) | |
| Conductor: | \(7581\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7581.cr
\(\chi_{7581}(20,\cdot)\) \(\chi_{7581}(419,\cdot)\) \(\chi_{7581}(818,\cdot)\) \(\chi_{7581}(1217,\cdot)\) \(\chi_{7581}(1616,\cdot)\) \(\chi_{7581}(2015,\cdot)\) \(\chi_{7581}(2414,\cdot)\) \(\chi_{7581}(2813,\cdot)\) \(\chi_{7581}(3212,\cdot)\) \(\chi_{7581}(4010,\cdot)\) \(\chi_{7581}(4409,\cdot)\) \(\chi_{7581}(4808,\cdot)\) \(\chi_{7581}(5207,\cdot)\) \(\chi_{7581}(5606,\cdot)\) \(\chi_{7581}(6005,\cdot)\) \(\chi_{7581}(6404,\cdot)\) \(\chi_{7581}(6803,\cdot)\) \(\chi_{7581}(7202,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
| Fixed field: | Number field defined by a degree 38 polynomial |
Values on generators
\((2528,6499,1807)\) → \((-1,-1,e\left(\frac{13}{19}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(20\) |
| \( \chi_{ 7581 }(20, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) |
sage: chi.jacobi_sum(n)