from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9225, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,22,3]))
pari: [g,chi] = znchar(Mod(298,9225))
Basic properties
Modulus: | \(9225\) | |
Conductor: | \(1025\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1025}(298,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9225.iq
\(\chi_{9225}(28,\cdot)\) \(\chi_{9225}(298,\cdot)\) \(\chi_{9225}(352,\cdot)\) \(\chi_{9225}(712,\cdot)\) \(\chi_{9225}(973,\cdot)\) \(\chi_{9225}(1072,\cdot)\) \(\chi_{9225}(3502,\cdot)\) \(\chi_{9225}(3538,\cdot)\) \(\chi_{9225}(4708,\cdot)\) \(\chi_{9225}(4942,\cdot)\) \(\chi_{9225}(4987,\cdot)\) \(\chi_{9225}(5113,\cdot)\) \(\chi_{9225}(6292,\cdot)\) \(\chi_{9225}(6472,\cdot)\) \(\chi_{9225}(6778,\cdot)\) \(\chi_{9225}(7633,\cdot)\)
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: | Number field defined by a degree 40 polynomial |
Values on generators
\((8201,1477,6976)\) → \((1,e\left(\frac{11}{20}\right),e\left(\frac{3}{40}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 9225 }(298, a) \) | \(1\) | \(1\) | \(-1\) | \(1\) | \(e\left(\frac{27}{40}\right)\) | \(-1\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) |
sage: chi.jacobi_sum(n)