from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,9]))
pari: [g,chi] = znchar(Mod(1126,2009))
Basic properties
Modulus: | \(2009\) | |
Conductor: | \(287\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{287}(265,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2009.bj
\(\chi_{2009}(48,\cdot)\) \(\chi_{2009}(97,\cdot)\) \(\chi_{2009}(293,\cdot)\) \(\chi_{2009}(391,\cdot)\) \(\chi_{2009}(440,\cdot)\) \(\chi_{2009}(587,\cdot)\) \(\chi_{2009}(685,\cdot)\) \(\chi_{2009}(832,\cdot)\) \(\chi_{2009}(930,\cdot)\) \(\chi_{2009}(1077,\cdot)\) \(\chi_{2009}(1126,\cdot)\) \(\chi_{2009}(1224,\cdot)\) \(\chi_{2009}(1420,\cdot)\) \(\chi_{2009}(1469,\cdot)\) \(\chi_{2009}(1616,\cdot)\) \(\chi_{2009}(1910,\cdot)\)
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: | 40.40.63172957949423116502957480067191906200305068755882825968063357506461803384975161.1 |
Values on generators
\((493,785)\) → \((-1,e\left(\frac{9}{40}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
\( \chi_{ 2009 }(1126, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(-i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) |
sage: chi.jacobi_sum(n)