from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,0,33,28]))
pari: [g,chi] = znchar(Mod(23,4840))
Basic properties
| Modulus: | \(4840\) | |
| Conductor: | \(2420\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2420}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4840.cm
\(\chi_{4840}(23,\cdot)\) \(\chi_{4840}(287,\cdot)\) \(\chi_{4840}(463,\cdot)\) \(\chi_{4840}(903,\cdot)\) \(\chi_{4840}(1167,\cdot)\) \(\chi_{4840}(1343,\cdot)\) \(\chi_{4840}(1607,\cdot)\) \(\chi_{4840}(1783,\cdot)\) \(\chi_{4840}(2047,\cdot)\) \(\chi_{4840}(2223,\cdot)\) \(\chi_{4840}(2487,\cdot)\) \(\chi_{4840}(2927,\cdot)\) \(\chi_{4840}(3103,\cdot)\) \(\chi_{4840}(3367,\cdot)\) \(\chi_{4840}(3543,\cdot)\) \(\chi_{4840}(3807,\cdot)\) \(\chi_{4840}(3983,\cdot)\) \(\chi_{4840}(4247,\cdot)\) \(\chi_{4840}(4423,\cdot)\) \(\chi_{4840}(4687,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((3631,2421,1937,4721)\) → \((-1,1,-i,e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 4840 }(23, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{31}{44}\right)\) | \(-1\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(i\) | \(e\left(\frac{7}{22}\right)\) |
sage: chi.jacobi_sum(n)