from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3960, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,50,45,48]))
pari: [g,chi] = znchar(Mod(113,3960))
Basic properties
| Modulus: | \(3960\) | |
| Conductor: | \(495\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{495}(113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3960.hi
\(\chi_{3960}(113,\cdot)\) \(\chi_{3960}(137,\cdot)\) \(\chi_{3960}(257,\cdot)\) \(\chi_{3960}(713,\cdot)\) \(\chi_{3960}(977,\cdot)\) \(\chi_{3960}(1193,\cdot)\) \(\chi_{3960}(1433,\cdot)\) \(\chi_{3960}(1577,\cdot)\) \(\chi_{3960}(1697,\cdot)\) \(\chi_{3960}(2297,\cdot)\) \(\chi_{3960}(2513,\cdot)\) \(\chi_{3960}(2633,\cdot)\) \(\chi_{3960}(2777,\cdot)\) \(\chi_{3960}(3017,\cdot)\) \(\chi_{3960}(3353,\cdot)\) \(\chi_{3960}(3953,\cdot)\)
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
\((991,1981,3521,2377,2521)\) → \((1,1,e\left(\frac{5}{6}\right),-i,e\left(\frac{4}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3960 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)