from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8008, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,50,48,35]))
pari: [g,chi] = znchar(Mod(817,8008))
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 \(\chi_{1001}(817,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8008.qj
\(\chi_{8008}(817,\cdot)\) \(\chi_{8008}(873,\cdot)\) \(\chi_{8008}(1137,\cdot)\) \(\chi_{8008}(1545,\cdot)\) \(\chi_{8008}(1697,\cdot)\) \(\chi_{8008}(2425,\cdot)\) \(\chi_{8008}(3001,\cdot)\) \(\chi_{8008}(3881,\cdot)\) \(\chi_{8008}(4513,\cdot)\) \(\chi_{8008}(4777,\cdot)\) \(\chi_{8008}(5185,\cdot)\) \(\chi_{8008}(5241,\cdot)\) \(\chi_{8008}(5505,\cdot)\) \(\chi_{8008}(6065,\cdot)\) \(\chi_{8008}(6697,\cdot)\) \(\chi_{8008}(6961,\cdot)\)
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
\((6007,4005,3433,4369,4929)\) → \((1,1,e\left(\frac{5}{6}\right),e\left(\frac{4}{5}\right),e\left(\frac{7}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
\( \chi_{ 8008 }(817, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(-1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) |
sage: chi.jacobi_sum(n)