from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8008, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,0,10,3,10]))
pari: [g,chi] = znchar(Mod(1927,8008))
Basic properties
Modulus: | \(8008\) | |
Conductor: | \(4004\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{4004}(1927,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8008.mz
\(\chi_{8008}(711,\cdot)\) \(\chi_{8008}(1927,\cdot)\) \(\chi_{8008}(2895,\cdot)\) \(\chi_{8008}(3383,\cdot)\) \(\chi_{8008}(4111,\cdot)\) \(\chi_{8008}(4351,\cdot)\) \(\chi_{8008}(5079,\cdot)\) \(\chi_{8008}(7751,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((6007,4005,3433,4369,4929)\) → \((-1,1,e\left(\frac{1}{3}\right),e\left(\frac{1}{10}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
\( \chi_{ 8008 }(1927, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(-1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{30}\right)\) |
sage: chi.jacobi_sum(n)