from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2310, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,40,36]))
pari: [g,chi] = znchar(Mod(53,2310))
Basic properties
| Modulus: | \(2310\) | |
| Conductor: | \(1155\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1155}(53,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2310.dl
\(\chi_{2310}(53,\cdot)\) \(\chi_{2310}(137,\cdot)\) \(\chi_{2310}(317,\cdot)\) \(\chi_{2310}(443,\cdot)\) \(\chi_{2310}(653,\cdot)\) \(\chi_{2310}(863,\cdot)\) \(\chi_{2310}(977,\cdot)\) \(\chi_{2310}(1103,\cdot)\) \(\chi_{2310}(1313,\cdot)\) \(\chi_{2310}(1367,\cdot)\) \(\chi_{2310}(1523,\cdot)\) \(\chi_{2310}(1577,\cdot)\) \(\chi_{2310}(1703,\cdot)\) \(\chi_{2310}(1787,\cdot)\) \(\chi_{2310}(2027,\cdot)\) \(\chi_{2310}(2237,\cdot)\)
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
\((1541,1387,661,211)\) → \((-1,-i,e\left(\frac{2}{3}\right),e\left(\frac{3}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 2310 }(53, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(i\) | \(e\left(\frac{23}{60}\right)\) |
sage: chi.jacobi_sum(n)