from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9200, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,10]))
pari: [g,chi] = znchar(Mod(2401,9200))
Basic properties
Modulus: | \(9200\) | |
Conductor: | \(23\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{23}(9,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9200.ce
\(\chi_{9200}(2401,\cdot)\) \(\chi_{9200}(2801,\cdot)\) \(\chi_{9200}(3201,\cdot)\) \(\chi_{9200}(3601,\cdot)\) \(\chi_{9200}(4401,\cdot)\) \(\chi_{9200}(5201,\cdot)\) \(\chi_{9200}(5601,\cdot)\) \(\chi_{9200}(6801,\cdot)\) \(\chi_{9200}(7201,\cdot)\) \(\chi_{9200}(8401,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
Fixed field: | \(\Q(\zeta_{23})^+\) |
Values on generators
\((1151,6901,2577,1201)\) → \((1,1,1,e\left(\frac{5}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 9200 }(2401, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) |
sage: chi.jacobi_sum(n)