from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(573, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,1]))
pari: [g,chi] = znchar(Mod(548,573))
Basic properties
| Modulus: | \(573\) | |
| Conductor: | \(573\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | 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 573.k
\(\chi_{573}(11,\cdot)\) \(\chi_{573}(14,\cdot)\) \(\chi_{573}(38,\cdot)\) \(\chi_{573}(41,\cdot)\) \(\chi_{573}(122,\cdot)\) \(\chi_{573}(155,\cdot)\) \(\chi_{573}(161,\cdot)\) \(\chi_{573}(185,\cdot)\) \(\chi_{573}(257,\cdot)\) \(\chi_{573}(275,\cdot)\) \(\chi_{573}(350,\cdot)\) \(\chi_{573}(377,\cdot)\) \(\chi_{573}(413,\cdot)\) \(\chi_{573}(419,\cdot)\) \(\chi_{573}(437,\cdot)\) \(\chi_{573}(452,\cdot)\) \(\chi_{573}(521,\cdot)\) \(\chi_{573}(548,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
| Fixed field: | 38.38.2907625224541948887418190194159474648332884150922805055774449275194637293139188364462591928677.1 |
Values on generators
\((383,19)\) → \((-1,e\left(\frac{1}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 573 }(548, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(-1\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) |
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)