from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3020, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,25,2]))
pari: [g,chi] = znchar(Mod(1809,3020))
Basic properties
Modulus: | \(3020\) | |
Conductor: | \(755\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{755}(299,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3020.bv
\(\chi_{3020}(9,\cdot)\) \(\chi_{3020}(29,\cdot)\) \(\chi_{3020}(229,\cdot)\) \(\chi_{3020}(249,\cdot)\) \(\chi_{3020}(429,\cdot)\) \(\chi_{3020}(729,\cdot)\) \(\chi_{3020}(849,\cdot)\) \(\chi_{3020}(1029,\cdot)\) \(\chi_{3020}(1129,\cdot)\) \(\chi_{3020}(1289,\cdot)\) \(\chi_{3020}(1409,\cdot)\) \(\chi_{3020}(1469,\cdot)\) \(\chi_{3020}(1729,\cdot)\) \(\chi_{3020}(1809,\cdot)\) \(\chi_{3020}(2049,\cdot)\) \(\chi_{3020}(2309,\cdot)\) \(\chi_{3020}(2349,\cdot)\) \(\chi_{3020}(2389,\cdot)\) \(\chi_{3020}(2809,\cdot)\) \(\chi_{3020}(2889,\cdot)\)
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
\((1511,2417,761)\) → \((1,-1,e\left(\frac{1}{25}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 3020 }(1809, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{50}\right)\) |
sage: chi.jacobi_sum(n)