from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,33,18]))
pari: [g,chi] = znchar(Mod(1253,1840))
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.cf
\(\chi_{1840}(53,\cdot)\) \(\chi_{1840}(157,\cdot)\) \(\chi_{1840}(237,\cdot)\) \(\chi_{1840}(293,\cdot)\) \(\chi_{1840}(373,\cdot)\) \(\chi_{1840}(477,\cdot)\) \(\chi_{1840}(557,\cdot)\) \(\chi_{1840}(613,\cdot)\) \(\chi_{1840}(773,\cdot)\) \(\chi_{1840}(797,\cdot)\) \(\chi_{1840}(957,\cdot)\) \(\chi_{1840}(1253,\cdot)\) \(\chi_{1840}(1413,\cdot)\) \(\chi_{1840}(1437,\cdot)\) \(\chi_{1840}(1493,\cdot)\) \(\chi_{1840}(1597,\cdot)\) \(\chi_{1840}(1653,\cdot)\) \(\chi_{1840}(1677,\cdot)\) \(\chi_{1840}(1813,\cdot)\) \(\chi_{1840}(1837,\cdot)\)
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
\((1151,1381,737,1201)\) → \((1,i,-i,e\left(\frac{9}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 1840 }(1253, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) |
sage: chi.jacobi_sum(n)