from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2400, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,35,0,38]))
pari: [g,chi] = znchar(Mod(13,2400))
Basic properties
| Modulus: | \(2400\) | |
| Conductor: | \(800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{800}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2400.du
\(\chi_{2400}(13,\cdot)\) \(\chi_{2400}(37,\cdot)\) \(\chi_{2400}(253,\cdot)\) \(\chi_{2400}(277,\cdot)\) \(\chi_{2400}(517,\cdot)\) \(\chi_{2400}(733,\cdot)\) \(\chi_{2400}(973,\cdot)\) \(\chi_{2400}(997,\cdot)\) \(\chi_{2400}(1213,\cdot)\) \(\chi_{2400}(1237,\cdot)\) \(\chi_{2400}(1453,\cdot)\) \(\chi_{2400}(1477,\cdot)\) \(\chi_{2400}(1717,\cdot)\) \(\chi_{2400}(1933,\cdot)\) \(\chi_{2400}(2173,\cdot)\) \(\chi_{2400}(2197,\cdot)\)
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: | 40.0.386856262276681335905976320000000000000000000000000000000000000000000000000000000000000000000000.1 |
Values on generators
\((1951,901,1601,577)\) → \((1,e\left(\frac{7}{8}\right),1,e\left(\frac{19}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2400 }(13, a) \) | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) |
sage: chi.jacobi_sum(n)