from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8325, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,18,35]))
pari: [g,chi] = znchar(Mod(3914,8325))
Basic properties
| Modulus: | \(8325\) | |
| Conductor: | \(2775\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2775}(1139,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8325.jh
\(\chi_{8325}(134,\cdot)\) \(\chi_{8325}(584,\cdot)\) \(\chi_{8325}(2339,\cdot)\) \(\chi_{8325}(2789,\cdot)\) \(\chi_{8325}(3464,\cdot)\) \(\chi_{8325}(3914,\cdot)\) \(\chi_{8325}(4004,\cdot)\) \(\chi_{8325}(4454,\cdot)\) \(\chi_{8325}(5129,\cdot)\) \(\chi_{8325}(5579,\cdot)\) \(\chi_{8325}(5669,\cdot)\) \(\chi_{8325}(6119,\cdot)\) \(\chi_{8325}(6794,\cdot)\) \(\chi_{8325}(7244,\cdot)\) \(\chi_{8325}(7334,\cdot)\) \(\chi_{8325}(7784,\cdot)\)
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
\((3701,7327,5626)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 8325 }(3914, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) |
sage: chi.jacobi_sum(n)