from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5400, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,40,27]))
pari: [g,chi] = znchar(Mod(937,5400))
Basic properties
| Modulus: | \(5400\) | |
| Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{225}(187,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5400.ep
\(\chi_{5400}(73,\cdot)\) \(\chi_{5400}(577,\cdot)\) \(\chi_{5400}(937,\cdot)\) \(\chi_{5400}(1153,\cdot)\) \(\chi_{5400}(1873,\cdot)\) \(\chi_{5400}(2017,\cdot)\) \(\chi_{5400}(2233,\cdot)\) \(\chi_{5400}(2737,\cdot)\) \(\chi_{5400}(2953,\cdot)\) \(\chi_{5400}(3097,\cdot)\) \(\chi_{5400}(3313,\cdot)\) \(\chi_{5400}(3817,\cdot)\) \(\chi_{5400}(4033,\cdot)\) \(\chi_{5400}(4177,\cdot)\) \(\chi_{5400}(4897,\cdot)\) \(\chi_{5400}(5113,\cdot)\)
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
\((1351,2701,1001,2377)\) → \((1,1,e\left(\frac{2}{3}\right),e\left(\frac{9}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5400 }(937, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) |
sage: chi.jacobi_sum(n)