from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1716, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,6,35]))
pari: [g,chi] = znchar(Mod(167,1716))
Basic properties
| Modulus: | \(1716\) | |
| Conductor: | \(1716\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1716.dm
\(\chi_{1716}(167,\cdot)\) \(\chi_{1716}(215,\cdot)\) \(\chi_{1716}(227,\cdot)\) \(\chi_{1716}(371,\cdot)\) \(\chi_{1716}(431,\cdot)\) \(\chi_{1716}(479,\cdot)\) \(\chi_{1716}(635,\cdot)\) \(\chi_{1716}(695,\cdot)\) \(\chi_{1716}(743,\cdot)\) \(\chi_{1716}(899,\cdot)\) \(\chi_{1716}(1007,\cdot)\) \(\chi_{1716}(1151,\cdot)\) \(\chi_{1716}(1163,\cdot)\) \(\chi_{1716}(1415,\cdot)\) \(\chi_{1716}(1619,\cdot)\) \(\chi_{1716}(1679,\cdot)\)
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
\((859,1145,937,925)\) → \((-1,-1,e\left(\frac{1}{10}\right),e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 1716 }(167, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) |
sage: chi.jacobi_sum(n)