from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4200, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,30,30,21,40]))
pari: [g,chi] = znchar(Mod(53,4200))
Basic properties
Modulus: | \(4200\) | |
Conductor: | \(4200\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4200.hj
\(\chi_{4200}(53,\cdot)\) \(\chi_{4200}(317,\cdot)\) \(\chi_{4200}(653,\cdot)\) \(\chi_{4200}(1397,\cdot)\) \(\chi_{4200}(1733,\cdot)\) \(\chi_{4200}(1997,\cdot)\) \(\chi_{4200}(2237,\cdot)\) \(\chi_{4200}(2333,\cdot)\) \(\chi_{4200}(2573,\cdot)\) \(\chi_{4200}(2837,\cdot)\) \(\chi_{4200}(3077,\cdot)\) \(\chi_{4200}(3173,\cdot)\) \(\chi_{4200}(3413,\cdot)\) \(\chi_{4200}(3677,\cdot)\) \(\chi_{4200}(3917,\cdot)\) \(\chi_{4200}(4013,\cdot)\)
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
\((3151,2101,2801,1177,3601)\) → \((1,-1,-1,e\left(\frac{7}{20}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 4200 }(53, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)