from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1968, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,20,0,29]))
pari: [g,chi] = znchar(Mod(391,1968))
Basic properties
| Modulus: | \(1968\) | |
| Conductor: | \(328\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{328}(227,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1968.dx
\(\chi_{1968}(7,\cdot)\) \(\chi_{1968}(151,\cdot)\) \(\chi_{1968}(199,\cdot)\) \(\chi_{1968}(343,\cdot)\) \(\chi_{1968}(391,\cdot)\) \(\chi_{1968}(439,\cdot)\) \(\chi_{1968}(727,\cdot)\) \(\chi_{1968}(919,\cdot)\) \(\chi_{1968}(967,\cdot)\) \(\chi_{1968}(1159,\cdot)\) \(\chi_{1968}(1447,\cdot)\) \(\chi_{1968}(1495,\cdot)\) \(\chi_{1968}(1543,\cdot)\) \(\chi_{1968}(1687,\cdot)\) \(\chi_{1968}(1735,\cdot)\) \(\chi_{1968}(1879,\cdot)\)
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: | 40.40.912788483257978497757884926199917783690257306123427760963531833190283833440731136.1 |
Values on generators
\((1231,1477,1313,1441)\) → \((-1,-1,1,e\left(\frac{29}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1968 }(391, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) |
sage: chi.jacobi_sum(n)