from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4005, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,42]))
pari: [g,chi] = znchar(Mod(278,4005))
Basic properties
Modulus: | \(4005\) | |
Conductor: | \(1335\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1335}(278,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4005.cu
\(\chi_{4005}(278,\cdot)\) \(\chi_{4005}(413,\cdot)\) \(\chi_{4005}(467,\cdot)\) \(\chi_{4005}(737,\cdot)\) \(\chi_{4005}(1052,\cdot)\) \(\chi_{4005}(1268,\cdot)\) \(\chi_{4005}(1538,\cdot)\) \(\chi_{4005}(1772,\cdot)\) \(\chi_{4005}(1853,\cdot)\) \(\chi_{4005}(2132,\cdot)\) \(\chi_{4005}(2312,\cdot)\) \(\chi_{4005}(2447,\cdot)\) \(\chi_{4005}(2573,\cdot)\) \(\chi_{4005}(2933,\cdot)\) \(\chi_{4005}(2987,\cdot)\) \(\chi_{4005}(3113,\cdot)\) \(\chi_{4005}(3248,\cdot)\) \(\chi_{4005}(3482,\cdot)\) \(\chi_{4005}(3617,\cdot)\) \(\chi_{4005}(3788,\cdot)\)
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
\((3116,802,181)\) → \((-1,-i,e\left(\frac{21}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 4005 }(278, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) |
sage: chi.jacobi_sum(n)