from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8370, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,15,14]))
pari: [g,chi] = znchar(Mod(17,8370))
Basic properties
| Modulus: | \(8370\) | |
| Conductor: | \(1395\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1395}(482,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8370.fn
\(\chi_{8370}(17,\cdot)\) \(\chi_{8370}(197,\cdot)\) \(\chi_{8370}(1097,\cdot)\) \(\chi_{8370}(1367,\cdot)\) \(\chi_{8370}(1853,\cdot)\) \(\chi_{8370}(2843,\cdot)\) \(\chi_{8370}(3113,\cdot)\) \(\chi_{8370}(3527,\cdot)\) \(\chi_{8370}(4517,\cdot)\) \(\chi_{8370}(4733,\cdot)\) \(\chi_{8370}(4787,\cdot)\) \(\chi_{8370}(6407,\cdot)\) \(\chi_{8370}(6713,\cdot)\) \(\chi_{8370}(6893,\cdot)\) \(\chi_{8370}(7793,\cdot)\) \(\chi_{8370}(8063,\cdot)\)
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
\((7751,6697,4591)\) → \((e\left(\frac{5}{6}\right),i,e\left(\frac{7}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8370 }(17, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) |
sage: chi.jacobi_sum(n)