from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4000, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,0,44]))
pari: [g,chi] = znchar(Mod(2431,4000))
Basic properties
Modulus: | \(4000\) | |
Conductor: | \(500\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{500}(431,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4000.cl
\(\chi_{4000}(31,\cdot)\) \(\chi_{4000}(191,\cdot)\) \(\chi_{4000}(511,\cdot)\) \(\chi_{4000}(671,\cdot)\) \(\chi_{4000}(831,\cdot)\) \(\chi_{4000}(991,\cdot)\) \(\chi_{4000}(1311,\cdot)\) \(\chi_{4000}(1471,\cdot)\) \(\chi_{4000}(1631,\cdot)\) \(\chi_{4000}(1791,\cdot)\) \(\chi_{4000}(2111,\cdot)\) \(\chi_{4000}(2271,\cdot)\) \(\chi_{4000}(2431,\cdot)\) \(\chi_{4000}(2591,\cdot)\) \(\chi_{4000}(2911,\cdot)\) \(\chi_{4000}(3071,\cdot)\) \(\chi_{4000}(3231,\cdot)\) \(\chi_{4000}(3391,\cdot)\) \(\chi_{4000}(3711,\cdot)\) \(\chi_{4000}(3871,\cdot)\)
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 50 polynomial |
Values on generators
\((2751,2501,1377)\) → \((-1,1,e\left(\frac{22}{25}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 4000 }(2431, a) \) | \(-1\) | \(1\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{49}{50}\right)\) |
sage: chi.jacobi_sum(n)