from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6030, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,11,12]))
pari: [g,chi] = znchar(Mod(107,6030))
Basic properties
| Modulus: | \(6030\) | |
| Conductor: | \(1005\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1005}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6030.cx
\(\chi_{6030}(107,\cdot)\) \(\chi_{6030}(143,\cdot)\) \(\chi_{6030}(863,\cdot)\) \(\chi_{6030}(953,\cdot)\) \(\chi_{6030}(1097,\cdot)\) \(\chi_{6030}(1313,\cdot)\) \(\chi_{6030}(2303,\cdot)\) \(\chi_{6030}(2627,\cdot)\) \(\chi_{6030}(3077,\cdot)\) \(\chi_{6030}(3347,\cdot)\) \(\chi_{6030}(3707,\cdot)\) \(\chi_{6030}(3833,\cdot)\) \(\chi_{6030}(3977,\cdot)\) \(\chi_{6030}(4283,\cdot)\) \(\chi_{6030}(4553,\cdot)\) \(\chi_{6030}(4913,\cdot)\) \(\chi_{6030}(4967,\cdot)\) \(\chi_{6030}(5183,\cdot)\) \(\chi_{6030}(5687,\cdot)\) \(\chi_{6030}(5777,\cdot)\)
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
\((4691,1207,3151)\) → \((-1,i,e\left(\frac{3}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6030 }(107, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(i\) | \(e\left(\frac{21}{22}\right)\) |
sage: chi.jacobi_sum(n)