from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8036, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,0,33]))
pari: [g,chi] = znchar(Mod(99,8036))
Basic properties
| Modulus: | \(8036\) | |
| Conductor: | \(164\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{164}(99,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8036.co
\(\chi_{8036}(99,\cdot)\) \(\chi_{8036}(883,\cdot)\) \(\chi_{8036}(1079,\cdot)\) \(\chi_{8036}(3431,\cdot)\) \(\chi_{8036}(3627,\cdot)\) \(\chi_{8036}(4411,\cdot)\) \(\chi_{8036}(4607,\cdot)\) \(\chi_{8036}(4803,\cdot)\) \(\chi_{8036}(5195,\cdot)\) \(\chi_{8036}(5587,\cdot)\) \(\chi_{8036}(5979,\cdot)\) \(\chi_{8036}(6567,\cdot)\) \(\chi_{8036}(6959,\cdot)\) \(\chi_{8036}(7351,\cdot)\) \(\chi_{8036}(7743,\cdot)\) \(\chi_{8036}(7939,\cdot)\)
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: | \(\Q(\zeta_{164})^+\) |
Values on generators
\((4019,493,785)\) → \((-1,1,e\left(\frac{33}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 8036 }(99, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(-i\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)