from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,11,4]))
pari: [g,chi] = znchar(Mod(1267,1840))
Basic properties
Modulus: | \(1840\) | |
Conductor: | \(1840\) | 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 1840.cy
\(\chi_{1840}(123,\cdot)\) \(\chi_{1840}(147,\cdot)\) \(\chi_{1840}(307,\cdot)\) \(\chi_{1840}(363,\cdot)\) \(\chi_{1840}(443,\cdot)\) \(\chi_{1840}(547,\cdot)\) \(\chi_{1840}(627,\cdot)\) \(\chi_{1840}(683,\cdot)\) \(\chi_{1840}(763,\cdot)\) \(\chi_{1840}(867,\cdot)\) \(\chi_{1840}(923,\cdot)\) \(\chi_{1840}(947,\cdot)\) \(\chi_{1840}(1083,\cdot)\) \(\chi_{1840}(1107,\cdot)\) \(\chi_{1840}(1163,\cdot)\) \(\chi_{1840}(1267,\cdot)\) \(\chi_{1840}(1323,\cdot)\) \(\chi_{1840}(1347,\cdot)\) \(\chi_{1840}(1507,\cdot)\) \(\chi_{1840}(1803,\cdot)\)
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
\((1151,1381,737,1201)\) → \((-1,-i,i,e\left(\frac{1}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 1840 }(1267, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) |
sage: chi.jacobi_sum(n)