from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4140, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,11,18]))
pari: [g,chi] = znchar(Mod(2357,4140))
Basic properties
| Modulus: | \(4140\) | |
| Conductor: | \(345\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{345}(287,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4140.cn
\(\chi_{4140}(17,\cdot)\) \(\chi_{4140}(53,\cdot)\) \(\chi_{4140}(557,\cdot)\) \(\chi_{4140}(773,\cdot)\) \(\chi_{4140}(917,\cdot)\) \(\chi_{4140}(953,\cdot)\) \(\chi_{4140}(1493,\cdot)\) \(\chi_{4140}(1673,\cdot)\) \(\chi_{4140}(1997,\cdot)\) \(\chi_{4140}(2177,\cdot)\) \(\chi_{4140}(2213,\cdot)\) \(\chi_{4140}(2357,\cdot)\) \(\chi_{4140}(2537,\cdot)\) \(\chi_{4140}(2573,\cdot)\) \(\chi_{4140}(3257,\cdot)\) \(\chi_{4140}(3437,\cdot)\) \(\chi_{4140}(3653,\cdot)\) \(\chi_{4140}(3833,\cdot)\) \(\chi_{4140}(3977,\cdot)\) \(\chi_{4140}(4013,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((2071,461,1657,3961)\) → \((1,-1,i,e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4140 }(2357, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{35}{44}\right)\) |
sage: chi.jacobi_sum(n)