from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,11,22,34]))
pari: [g,chi] = znchar(Mod(1579,1840))
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.cj
\(\chi_{1840}(19,\cdot)\) \(\chi_{1840}(99,\cdot)\) \(\chi_{1840}(339,\cdot)\) \(\chi_{1840}(379,\cdot)\) \(\chi_{1840}(419,\cdot)\) \(\chi_{1840}(539,\cdot)\) \(\chi_{1840}(619,\cdot)\) \(\chi_{1840}(659,\cdot)\) \(\chi_{1840}(779,\cdot)\) \(\chi_{1840}(819,\cdot)\) \(\chi_{1840}(939,\cdot)\) \(\chi_{1840}(1019,\cdot)\) \(\chi_{1840}(1259,\cdot)\) \(\chi_{1840}(1299,\cdot)\) \(\chi_{1840}(1339,\cdot)\) \(\chi_{1840}(1459,\cdot)\) \(\chi_{1840}(1539,\cdot)\) \(\chi_{1840}(1579,\cdot)\) \(\chi_{1840}(1699,\cdot)\) \(\chi_{1840}(1739,\cdot)\)
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
\((1151,1381,737,1201)\) → \((-1,i,-1,e\left(\frac{17}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 1840 }(1579, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) |
sage: chi.jacobi_sum(n)