from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,0,33,18]))
pari: [g,chi] = znchar(Mod(1713,1840))
Basic properties
Modulus: | \(1840\) | |
Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{115}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.ct
\(\chi_{1840}(17,\cdot)\) \(\chi_{1840}(33,\cdot)\) \(\chi_{1840}(97,\cdot)\) \(\chi_{1840}(113,\cdot)\) \(\chi_{1840}(273,\cdot)\) \(\chi_{1840}(337,\cdot)\) \(\chi_{1840}(433,\cdot)\) \(\chi_{1840}(497,\cdot)\) \(\chi_{1840}(513,\cdot)\) \(\chi_{1840}(753,\cdot)\) \(\chi_{1840}(833,\cdot)\) \(\chi_{1840}(977,\cdot)\) \(\chi_{1840}(1073,\cdot)\) \(\chi_{1840}(1137,\cdot)\) \(\chi_{1840}(1217,\cdot)\) \(\chi_{1840}(1233,\cdot)\) \(\chi_{1840}(1377,\cdot)\) \(\chi_{1840}(1537,\cdot)\) \(\chi_{1840}(1617,\cdot)\) \(\chi_{1840}(1713,\cdot)\)
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: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((1151,1381,737,1201)\) → \((1,1,-i,e\left(\frac{9}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 1840 }(1713, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) |
sage: chi.jacobi_sum(n)