from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,11,0,20]))
pari: [g,chi] = znchar(Mod(3137,4830))
Basic properties
| Modulus: | \(4830\) | |
| 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 \(\chi_{345}(32,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4830.ct
\(\chi_{4830}(197,\cdot)\) \(\chi_{4830}(407,\cdot)\) \(\chi_{4830}(533,\cdot)\) \(\chi_{4830}(1037,\cdot)\) \(\chi_{4830}(1163,\cdot)\) \(\chi_{4830}(1373,\cdot)\) \(\chi_{4830}(1457,\cdot)\) \(\chi_{4830}(2003,\cdot)\) \(\chi_{4830}(2423,\cdot)\) \(\chi_{4830}(2717,\cdot)\) \(\chi_{4830}(2927,\cdot)\) \(\chi_{4830}(3137,\cdot)\) \(\chi_{4830}(3347,\cdot)\) \(\chi_{4830}(3683,\cdot)\) \(\chi_{4830}(3767,\cdot)\) \(\chi_{4830}(3893,\cdot)\) \(\chi_{4830}(4103,\cdot)\) \(\chi_{4830}(4313,\cdot)\) \(\chi_{4830}(4397,\cdot)\) \(\chi_{4830}(4733,\cdot)\)
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
\((3221,967,2761,1891)\) → \((-1,i,1,e\left(\frac{5}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4830 }(3137, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)