from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2640, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,15,10,5,14]))
pari: [g,chi] = znchar(Mod(227,2640))
Basic properties
| Modulus: | \(2640\) | |
| Conductor: | \(2640\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | 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 2640.ft
\(\chi_{2640}(227,\cdot)\) \(\chi_{2640}(1163,\cdot)\) \(\chi_{2640}(1403,\cdot)\) \(\chi_{2640}(1427,\cdot)\) \(\chi_{2640}(1667,\cdot)\) \(\chi_{2640}(1883,\cdot)\) \(\chi_{2640}(2147,\cdot)\) \(\chi_{2640}(2603,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | 20.20.360977976896857923653306611918700544000000000000000.2 |
Values on generators
\((991,661,881,1057,1201)\) → \((-1,-i,-1,i,e\left(\frac{7}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2640 }(227, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(i\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)