from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10000, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,0,14]))
pari: [g,chi] = znchar(Mod(4401,10000))
Basic properties
| Modulus: | \(10000\) | |
| Conductor: | \(125\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(25\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{125}(81,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 10000.bo
\(\chi_{10000}(401,\cdot)\) \(\chi_{10000}(801,\cdot)\) \(\chi_{10000}(1201,\cdot)\) \(\chi_{10000}(1601,\cdot)\) \(\chi_{10000}(2401,\cdot)\) \(\chi_{10000}(2801,\cdot)\) \(\chi_{10000}(3201,\cdot)\) \(\chi_{10000}(3601,\cdot)\) \(\chi_{10000}(4401,\cdot)\) \(\chi_{10000}(4801,\cdot)\) \(\chi_{10000}(5201,\cdot)\) \(\chi_{10000}(5601,\cdot)\) \(\chi_{10000}(6401,\cdot)\) \(\chi_{10000}(6801,\cdot)\) \(\chi_{10000}(7201,\cdot)\) \(\chi_{10000}(7601,\cdot)\) \(\chi_{10000}(8401,\cdot)\) \(\chi_{10000}(8801,\cdot)\) \(\chi_{10000}(9201,\cdot)\) \(\chi_{10000}(9601,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 25 polynomial |
Values on generators
\((8751,2501,9377)\) → \((1,1,e\left(\frac{7}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 10000 }(4401, a) \) | \(1\) | \(1\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) |
sage: chi.jacobi_sum(n)