from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3040, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,0,26]))
pari: [g,chi] = znchar(Mod(41,3040))
Basic properties
| Modulus: | \(3040\) | |
| Conductor: | \(304\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{304}(269,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3040.fu
\(\chi_{3040}(41,\cdot)\) \(\chi_{3040}(281,\cdot)\) \(\chi_{3040}(1001,\cdot)\) \(\chi_{3040}(1161,\cdot)\) \(\chi_{3040}(1321,\cdot)\) \(\chi_{3040}(1401,\cdot)\) \(\chi_{3040}(1561,\cdot)\) \(\chi_{3040}(1801,\cdot)\) \(\chi_{3040}(2521,\cdot)\) \(\chi_{3040}(2681,\cdot)\) \(\chi_{3040}(2841,\cdot)\) \(\chi_{3040}(2921,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.0.19036714782161565107424425435655777110146017378670996611401194085493506048.1 |
Values on generators
\((191,2661,1217,1921)\) → \((1,-i,1,e\left(\frac{13}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3040 }(41, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{19}{36}\right)\) |
sage: chi.jacobi_sum(n)