from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9240, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,0,15,50,6]))
pari: [g,chi] = znchar(Mod(5227,9240))
Basic properties
Modulus: | \(9240\) | |
Conductor: | \(3080\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3080}(2147,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9240.ob
\(\chi_{9240}(283,\cdot)\) \(\chi_{9240}(523,\cdot)\) \(\chi_{9240}(787,\cdot)\) \(\chi_{9240}(1867,\cdot)\) \(\chi_{9240}(3043,\cdot)\) \(\chi_{9240}(3307,\cdot)\) \(\chi_{9240}(3643,\cdot)\) \(\chi_{9240}(4483,\cdot)\) \(\chi_{9240}(5227,\cdot)\) \(\chi_{9240}(5563,\cdot)\) \(\chi_{9240}(5827,\cdot)\) \(\chi_{9240}(6067,\cdot)\) \(\chi_{9240}(7003,\cdot)\) \(\chi_{9240}(8587,\cdot)\) \(\chi_{9240}(8923,\cdot)\) \(\chi_{9240}(9187,\cdot)\)
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
\((2311,4621,6161,3697,5281,2521)\) → \((-1,-1,1,i,e\left(\frac{5}{6}\right),e\left(\frac{1}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 9240 }(5227, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(i\) | \(e\left(\frac{43}{60}\right)\) |
sage: chi.jacobi_sum(n)