from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(328, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,20,31]))
pari: [g,chi] = znchar(Mod(13,328))
Basic properties
| Modulus: | \(328\) | |
| Conductor: | \(328\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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 328.bf
\(\chi_{328}(13,\cdot)\) \(\chi_{328}(29,\cdot)\) \(\chi_{328}(53,\cdot)\) \(\chi_{328}(69,\cdot)\) \(\chi_{328}(93,\cdot)\) \(\chi_{328}(101,\cdot)\) \(\chi_{328}(117,\cdot)\) \(\chi_{328}(149,\cdot)\) \(\chi_{328}(157,\cdot)\) \(\chi_{328}(181,\cdot)\) \(\chi_{328}(229,\cdot)\) \(\chi_{328}(253,\cdot)\) \(\chi_{328}(261,\cdot)\) \(\chi_{328}(293,\cdot)\) \(\chi_{328}(309,\cdot)\) \(\chi_{328}(317,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.0.912788483257978497757884926199917783690257306123427760963531833190283833440731136.1 |
Values on generators
\((247,165,129)\) → \((1,-1,e\left(\frac{31}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 328 }(13, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(i\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) |
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)