from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1104, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,11,22,4]))
pari: [g,chi] = znchar(Mod(347,1104))
Basic properties
Modulus: | \(1104\) | |
Conductor: | \(1104\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1104.br
\(\chi_{1104}(35,\cdot)\) \(\chi_{1104}(59,\cdot)\) \(\chi_{1104}(131,\cdot)\) \(\chi_{1104}(179,\cdot)\) \(\chi_{1104}(347,\cdot)\) \(\chi_{1104}(371,\cdot)\) \(\chi_{1104}(395,\cdot)\) \(\chi_{1104}(443,\cdot)\) \(\chi_{1104}(491,\cdot)\) \(\chi_{1104}(515,\cdot)\) \(\chi_{1104}(587,\cdot)\) \(\chi_{1104}(611,\cdot)\) \(\chi_{1104}(683,\cdot)\) \(\chi_{1104}(731,\cdot)\) \(\chi_{1104}(899,\cdot)\) \(\chi_{1104}(923,\cdot)\) \(\chi_{1104}(947,\cdot)\) \(\chi_{1104}(995,\cdot)\) \(\chi_{1104}(1043,\cdot)\) \(\chi_{1104}(1067,\cdot)\)
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
\((415,277,737,97)\) → \((-1,i,-1,e\left(\frac{1}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 1104 }(347, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{25}{44}\right)\) |
sage: chi.jacobi_sum(n)