from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9900, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,50,3,0]))
pari: [g,chi] = znchar(Mod(4577,9900))
Basic properties
Modulus: | \(9900\) | |
Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{225}(77,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9900.ms
\(\chi_{9900}(353,\cdot)\) \(\chi_{9900}(617,\cdot)\) \(\chi_{9900}(1013,\cdot)\) \(\chi_{9900}(1937,\cdot)\) \(\chi_{9900}(2333,\cdot)\) \(\chi_{9900}(2597,\cdot)\) \(\chi_{9900}(3917,\cdot)\) \(\chi_{9900}(4313,\cdot)\) \(\chi_{9900}(4577,\cdot)\) \(\chi_{9900}(4973,\cdot)\) \(\chi_{9900}(5897,\cdot)\) \(\chi_{9900}(6953,\cdot)\) \(\chi_{9900}(7877,\cdot)\) \(\chi_{9900}(8273,\cdot)\) \(\chi_{9900}(8537,\cdot)\) \(\chi_{9900}(8933,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((4951,5501,2377,4501)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{1}{20}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 9900 }(4577, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)