from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1232, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,5,0,18]))
pari: [g,chi] = znchar(Mod(1205,1232))
Basic properties
| Modulus: | \(1232\) | |
| Conductor: | \(176\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{176}(149,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1232.cp
\(\chi_{1232}(29,\cdot)\) \(\chi_{1232}(85,\cdot)\) \(\chi_{1232}(365,\cdot)\) \(\chi_{1232}(589,\cdot)\) \(\chi_{1232}(645,\cdot)\) \(\chi_{1232}(701,\cdot)\) \(\chi_{1232}(981,\cdot)\) \(\chi_{1232}(1205,\cdot)\)
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: | 20.0.200317132330035063121671003054276608.1 |
Values on generators
\((463,309,353,673)\) → \((1,i,1,e\left(\frac{9}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 1232 }(1205, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(-1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) |
sage: chi.jacobi_sum(n)