from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(920, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,33,26]))
pari: [g,chi] = znchar(Mod(573,920))
Basic properties
| Modulus: | \(920\) | |
| Conductor: | \(920\) | 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 920.bo
\(\chi_{920}(37,\cdot)\) \(\chi_{920}(53,\cdot)\) \(\chi_{920}(157,\cdot)\) \(\chi_{920}(237,\cdot)\) \(\chi_{920}(293,\cdot)\) \(\chi_{920}(333,\cdot)\) \(\chi_{920}(373,\cdot)\) \(\chi_{920}(477,\cdot)\) \(\chi_{920}(493,\cdot)\) \(\chi_{920}(517,\cdot)\) \(\chi_{920}(557,\cdot)\) \(\chi_{920}(573,\cdot)\) \(\chi_{920}(613,\cdot)\) \(\chi_{920}(677,\cdot)\) \(\chi_{920}(733,\cdot)\) \(\chi_{920}(757,\cdot)\) \(\chi_{920}(773,\cdot)\) \(\chi_{920}(797,\cdot)\) \(\chi_{920}(893,\cdot)\) \(\chi_{920}(917,\cdot)\)
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: | 44.44.13383169230192059253459701104387771124501004765020501667165784506368000000000000000000000000000000000.1 |
Values on generators
\((231,461,737,281)\) → \((1,-1,-i,e\left(\frac{13}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 920 }(573, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)