from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(310, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,2]))
pari: [g,chi] = znchar(Mod(3,310))
Basic properties
| Modulus: | \(310\) | |
| Conductor: | \(155\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{155}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 310.w
\(\chi_{310}(3,\cdot)\) \(\chi_{310}(13,\cdot)\) \(\chi_{310}(17,\cdot)\) \(\chi_{310}(43,\cdot)\) \(\chi_{310}(53,\cdot)\) \(\chi_{310}(73,\cdot)\) \(\chi_{310}(83,\cdot)\) \(\chi_{310}(117,\cdot)\) \(\chi_{310}(127,\cdot)\) \(\chi_{310}(137,\cdot)\) \(\chi_{310}(167,\cdot)\) \(\chi_{310}(177,\cdot)\) \(\chi_{310}(197,\cdot)\) \(\chi_{310}(203,\cdot)\) \(\chi_{310}(207,\cdot)\) \(\chi_{310}(303,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((187,251)\) → \((-i,e\left(\frac{1}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 310 }(3, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) |
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)