from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2310, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,50,12]))
pari: [g,chi] = znchar(Mod(103,2310))
Basic properties
| Modulus: | \(2310\) | |
| Conductor: | \(385\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{385}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2310.dq
\(\chi_{2310}(103,\cdot)\) \(\chi_{2310}(157,\cdot)\) \(\chi_{2310}(313,\cdot)\) \(\chi_{2310}(367,\cdot)\) \(\chi_{2310}(493,\cdot)\) \(\chi_{2310}(577,\cdot)\) \(\chi_{2310}(817,\cdot)\) \(\chi_{2310}(1027,\cdot)\) \(\chi_{2310}(1153,\cdot)\) \(\chi_{2310}(1237,\cdot)\) \(\chi_{2310}(1417,\cdot)\) \(\chi_{2310}(1543,\cdot)\) \(\chi_{2310}(1753,\cdot)\) \(\chi_{2310}(1963,\cdot)\) \(\chi_{2310}(2077,\cdot)\) \(\chi_{2310}(2203,\cdot)\)
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
\((1541,1387,661,211)\) → \((1,-i,e\left(\frac{5}{6}\right),e\left(\frac{1}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 2310 }(103, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(i\) | \(e\left(\frac{31}{60}\right)\) |
sage: chi.jacobi_sum(n)