from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1980, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,50,45,48]))
pari: [g,chi] = znchar(Mod(113,1980))
Basic properties
| Modulus: | \(1980\) | |
| 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 1980.dr
\(\chi_{1980}(113,\cdot)\) \(\chi_{1980}(137,\cdot)\) \(\chi_{1980}(257,\cdot)\) \(\chi_{1980}(317,\cdot)\) \(\chi_{1980}(533,\cdot)\) \(\chi_{1980}(653,\cdot)\) \(\chi_{1980}(713,\cdot)\) \(\chi_{1980}(797,\cdot)\) \(\chi_{1980}(977,\cdot)\) \(\chi_{1980}(1037,\cdot)\) \(\chi_{1980}(1193,\cdot)\) \(\chi_{1980}(1373,\cdot)\) \(\chi_{1980}(1433,\cdot)\) \(\chi_{1980}(1577,\cdot)\) \(\chi_{1980}(1697,\cdot)\) \(\chi_{1980}(1973,\cdot)\)
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
\((991,1541,397,541)\) → \((1,e\left(\frac{5}{6}\right),-i,e\left(\frac{4}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1980 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)