from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10000, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,0,47]))
pari: [g,chi] = znchar(Mod(49,10000))
Basic properties
Modulus: | \(10000\) | |
Conductor: | \(125\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{125}(84,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 10000.bu
\(\chi_{10000}(49,\cdot)\) \(\chi_{10000}(449,\cdot)\) \(\chi_{10000}(849,\cdot)\) \(\chi_{10000}(1649,\cdot)\) \(\chi_{10000}(2049,\cdot)\) \(\chi_{10000}(2449,\cdot)\) \(\chi_{10000}(2849,\cdot)\) \(\chi_{10000}(3649,\cdot)\) \(\chi_{10000}(4049,\cdot)\) \(\chi_{10000}(4449,\cdot)\) \(\chi_{10000}(4849,\cdot)\) \(\chi_{10000}(5649,\cdot)\) \(\chi_{10000}(6049,\cdot)\) \(\chi_{10000}(6449,\cdot)\) \(\chi_{10000}(6849,\cdot)\) \(\chi_{10000}(7649,\cdot)\) \(\chi_{10000}(8049,\cdot)\) \(\chi_{10000}(8449,\cdot)\) \(\chi_{10000}(8849,\cdot)\) \(\chi_{10000}(9649,\cdot)\)
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
\((8751,2501,9377)\) → \((1,1,e\left(\frac{47}{50}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 10000 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{37}{50}\right)\) |
sage: chi.jacobi_sum(n)