from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2310, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,5,0,18]))
pari: [g,chi] = znchar(Mod(127,2310))
Basic properties
| Modulus: | \(2310\) | |
| Conductor: | \(55\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{55}(17,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2310.cn
\(\chi_{2310}(127,\cdot)\) \(\chi_{2310}(337,\cdot)\) \(\chi_{2310}(547,\cdot)\) \(\chi_{2310}(673,\cdot)\) \(\chi_{2310}(1513,\cdot)\) \(\chi_{2310}(1597,\cdot)\) \(\chi_{2310}(1723,\cdot)\) \(\chi_{2310}(1933,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | \(\Q(\zeta_{55})^+\) |
Values on generators
\((1541,1387,661,211)\) → \((1,i,1,e\left(\frac{9}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 2310 }(127, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-i\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(e\left(\frac{9}{20}\right)\) |
sage: chi.jacobi_sum(n)