from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2760, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,0,11,42]))
pari: [g,chi] = znchar(Mod(37,2760))
Basic properties
| Modulus: | \(2760\) | |
| Conductor: | \(920\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{920}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2760.dg
\(\chi_{2760}(37,\cdot)\) \(\chi_{2760}(157,\cdot)\) \(\chi_{2760}(373,\cdot)\) \(\chi_{2760}(493,\cdot)\) \(\chi_{2760}(517,\cdot)\) \(\chi_{2760}(613,\cdot)\) \(\chi_{2760}(733,\cdot)\) \(\chi_{2760}(757,\cdot)\) \(\chi_{2760}(973,\cdot)\) \(\chi_{2760}(1213,\cdot)\) \(\chi_{2760}(1477,\cdot)\) \(\chi_{2760}(1597,\cdot)\) \(\chi_{2760}(1693,\cdot)\) \(\chi_{2760}(1717,\cdot)\) \(\chi_{2760}(1813,\cdot)\) \(\chi_{2760}(1837,\cdot)\) \(\chi_{2760}(2077,\cdot)\) \(\chi_{2760}(2173,\cdot)\) \(\chi_{2760}(2317,\cdot)\) \(\chi_{2760}(2413,\cdot)\)
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: | 44.44.13383169230192059253459701104387771124501004765020501667165784506368000000000000000000000000000000000.1 |
Values on generators
\((2071,1381,1841,1657,1201)\) → \((1,-1,1,i,e\left(\frac{21}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2760 }(37, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) |
sage: chi.jacobi_sum(n)