from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(750, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,17]))
pari: [g,chi] = znchar(Mod(59,750))
Basic properties
| Modulus: | \(750\) | |
| Conductor: | \(375\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{375}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 750.p
\(\chi_{750}(29,\cdot)\) \(\chi_{750}(59,\cdot)\) \(\chi_{750}(89,\cdot)\) \(\chi_{750}(119,\cdot)\) \(\chi_{750}(179,\cdot)\) \(\chi_{750}(209,\cdot)\) \(\chi_{750}(239,\cdot)\) \(\chi_{750}(269,\cdot)\) \(\chi_{750}(329,\cdot)\) \(\chi_{750}(359,\cdot)\) \(\chi_{750}(389,\cdot)\) \(\chi_{750}(419,\cdot)\) \(\chi_{750}(479,\cdot)\) \(\chi_{750}(509,\cdot)\) \(\chi_{750}(539,\cdot)\) \(\chi_{750}(569,\cdot)\) \(\chi_{750}(629,\cdot)\) \(\chi_{750}(659,\cdot)\) \(\chi_{750}(689,\cdot)\) \(\chi_{750}(719,\cdot)\)
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
\((251,127)\) → \((-1,e\left(\frac{17}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 750 }(59, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{23}{50}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)