from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,11,20]))
pari: [g,chi] = znchar(Mod(1757,1840))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.ce
\(\chi_{1840}(77,\cdot)\) \(\chi_{1840}(133,\cdot)\) \(\chi_{1840}(213,\cdot)\) \(\chi_{1840}(317,\cdot)\) \(\chi_{1840}(397,\cdot)\) \(\chi_{1840}(453,\cdot)\) \(\chi_{1840}(533,\cdot)\) \(\chi_{1840}(637,\cdot)\) \(\chi_{1840}(693,\cdot)\) \(\chi_{1840}(717,\cdot)\) \(\chi_{1840}(853,\cdot)\) \(\chi_{1840}(877,\cdot)\) \(\chi_{1840}(933,\cdot)\) \(\chi_{1840}(1037,\cdot)\) \(\chi_{1840}(1093,\cdot)\) \(\chi_{1840}(1117,\cdot)\) \(\chi_{1840}(1277,\cdot)\) \(\chi_{1840}(1573,\cdot)\) \(\chi_{1840}(1733,\cdot)\) \(\chi_{1840}(1757,\cdot)\)
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
\((1151,1381,737,1201)\) → \((1,-i,i,e\left(\frac{5}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 1840 }(1757, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) |
sage: chi.jacobi_sum(n)