from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,2]))
pari: [g,chi] = znchar(Mod(313,930))
Basic properties
Modulus: | \(930\) | |
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 930.bt
\(\chi_{930}(13,\cdot)\) \(\chi_{930}(43,\cdot)\) \(\chi_{930}(73,\cdot)\) \(\chi_{930}(127,\cdot)\) \(\chi_{930}(313,\cdot)\) \(\chi_{930}(427,\cdot)\) \(\chi_{930}(487,\cdot)\) \(\chi_{930}(517,\cdot)\) \(\chi_{930}(613,\cdot)\) \(\chi_{930}(637,\cdot)\) \(\chi_{930}(673,\cdot)\) \(\chi_{930}(703,\cdot)\) \(\chi_{930}(757,\cdot)\) \(\chi_{930}(787,\cdot)\) \(\chi_{930}(817,\cdot)\) \(\chi_{930}(823,\cdot)\)
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
\((311,187,871)\) → \((1,-i,e\left(\frac{1}{30}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 930 }(313, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) |
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)