from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9450, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,11,0]))
pari: [g,chi] = znchar(Mod(323,9450))
Basic properties
Modulus: | \(9450\) | |
Conductor: | \(75\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{75}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9450.dv
\(\chi_{9450}(323,\cdot)\) \(\chi_{9450}(2213,\cdot)\) \(\chi_{9450}(3347,\cdot)\) \(\chi_{9450}(4103,\cdot)\) \(\chi_{9450}(5237,\cdot)\) \(\chi_{9450}(7127,\cdot)\) \(\chi_{9450}(7883,\cdot)\) \(\chi_{9450}(9017,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | \(\Q(\zeta_{75})^+\) |
Values on generators
\((9101,6427,6751)\) → \((-1,e\left(\frac{11}{20}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 9450 }(323, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)