from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8008, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,40,6,45]))
pari: [g,chi] = znchar(Mod(1201,8008))
Basic properties
| Modulus: | \(8008\) | |
| Conductor: | \(1001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1001}(200,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8008.rx
\(\chi_{8008}(1201,\cdot)\) \(\chi_{8008}(1425,\cdot)\) \(\chi_{8008}(2153,\cdot)\) \(\chi_{8008}(2361,\cdot)\) \(\chi_{8008}(2657,\cdot)\) \(\chi_{8008}(3385,\cdot)\) \(\chi_{8008}(3593,\cdot)\) \(\chi_{8008}(4545,\cdot)\) \(\chi_{8008}(5777,\cdot)\) \(\chi_{8008}(5793,\cdot)\) \(\chi_{8008}(6001,\cdot)\) \(\chi_{8008}(6729,\cdot)\) \(\chi_{8008}(7025,\cdot)\) \(\chi_{8008}(7233,\cdot)\) \(\chi_{8008}(7961,\cdot)\) \(\chi_{8008}(7977,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((6007,4005,3433,4369,4929)\) → \((1,1,e\left(\frac{2}{3}\right),e\left(\frac{1}{10}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
| \( \chi_{ 8008 }(1201, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) |
sage: chi.jacobi_sum(n)