from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8040, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,0,22,11,16]))
pari: [g,chi] = znchar(Mod(617,8040))
Basic properties
| Modulus: | \(8040\) | |
| 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 \(\chi_{1005}(617,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8040.fn
\(\chi_{8040}(617,\cdot)\) \(\chi_{8040}(953,\cdot)\) \(\chi_{8040}(1097,\cdot)\) \(\chi_{8040}(1313,\cdot)\) \(\chi_{8040}(1337,\cdot)\) \(\chi_{8040}(1697,\cdot)\) \(\chi_{8040}(2153,\cdot)\) \(\chi_{8040}(2273,\cdot)\) \(\chi_{8040}(2873,\cdot)\) \(\chi_{8040}(3833,\cdot)\) \(\chi_{8040}(3977,\cdot)\) \(\chi_{8040}(4313,\cdot)\) \(\chi_{8040}(4553,\cdot)\) \(\chi_{8040}(4913,\cdot)\) \(\chi_{8040}(5777,\cdot)\) \(\chi_{8040}(6137,\cdot)\) \(\chi_{8040}(6977,\cdot)\) \(\chi_{8040}(7097,\cdot)\) \(\chi_{8040}(7193,\cdot)\) \(\chi_{8040}(7697,\cdot)\)
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
\((6031,4021,2681,3217,5161)\) → \((1,1,-1,i,e\left(\frac{4}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8040 }(617, a) \) | \(1\) | \(1\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(i\) | \(e\left(\frac{17}{22}\right)\) |
sage: chi.jacobi_sum(n)