from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3020, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,25,23]))
pari: [g,chi] = znchar(Mod(2819,3020))
Basic properties
| Modulus: | \(3020\) | |
| Conductor: | \(3020\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3020.bz
\(\chi_{3020}(79,\cdot)\) \(\chi_{3020}(179,\cdot)\) \(\chi_{3020}(359,\cdot)\) \(\chi_{3020}(479,\cdot)\) \(\chi_{3020}(779,\cdot)\) \(\chi_{3020}(959,\cdot)\) \(\chi_{3020}(979,\cdot)\) \(\chi_{3020}(1179,\cdot)\) \(\chi_{3020}(1199,\cdot)\) \(\chi_{3020}(1339,\cdot)\) \(\chi_{3020}(1419,\cdot)\) \(\chi_{3020}(1839,\cdot)\) \(\chi_{3020}(1879,\cdot)\) \(\chi_{3020}(1919,\cdot)\) \(\chi_{3020}(2179,\cdot)\) \(\chi_{3020}(2419,\cdot)\) \(\chi_{3020}(2499,\cdot)\) \(\chi_{3020}(2759,\cdot)\) \(\chi_{3020}(2819,\cdot)\) \(\chi_{3020}(2939,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 50 polynomial |
Values on generators
\((1511,2417,761)\) → \((-1,-1,e\left(\frac{23}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 3020 }(2819, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{39}{50}\right)\) |
sage: chi.jacobi_sum(n)