from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6825, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,3,20,55]))
pari: [g,chi] = znchar(Mod(5077,6825))
Basic properties
| Modulus: | \(6825\) | |
| Conductor: | \(2275\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2275}(527,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6825.ma
\(\chi_{6825}(58,\cdot)\) \(\chi_{6825}(613,\cdot)\) \(\chi_{6825}(1012,\cdot)\) \(\chi_{6825}(1423,\cdot)\) \(\chi_{6825}(1978,\cdot)\) \(\chi_{6825}(2347,\cdot)\) \(\chi_{6825}(2377,\cdot)\) \(\chi_{6825}(2788,\cdot)\) \(\chi_{6825}(3712,\cdot)\) \(\chi_{6825}(3742,\cdot)\) \(\chi_{6825}(4153,\cdot)\) \(\chi_{6825}(4708,\cdot)\) \(\chi_{6825}(5077,\cdot)\) \(\chi_{6825}(6073,\cdot)\) \(\chi_{6825}(6442,\cdot)\) \(\chi_{6825}(6472,\cdot)\)
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
\((2276,3277,976,4201)\) → \((1,e\left(\frac{1}{20}\right),e\left(\frac{1}{3}\right),e\left(\frac{11}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) | \(29\) |
| \( \chi_{ 6825 }(5077, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) |
sage: chi.jacobi_sum(n)