from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8008, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,30,20,6,15]))
pari: [g,chi] = znchar(Mod(541,8008))
Basic properties
| Modulus: | \(8008\) | |
| Conductor: | \(8008\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 8008.qy
\(\chi_{8008}(541,\cdot)\) \(\chi_{8008}(1773,\cdot)\) \(\chi_{8008}(1789,\cdot)\) \(\chi_{8008}(1997,\cdot)\) \(\chi_{8008}(2725,\cdot)\) \(\chi_{8008}(3021,\cdot)\) \(\chi_{8008}(3229,\cdot)\) \(\chi_{8008}(3957,\cdot)\) \(\chi_{8008}(3973,\cdot)\) \(\chi_{8008}(5205,\cdot)\) \(\chi_{8008}(5429,\cdot)\) \(\chi_{8008}(6157,\cdot)\) \(\chi_{8008}(6365,\cdot)\) \(\chi_{8008}(6661,\cdot)\) \(\chi_{8008}(7389,\cdot)\) \(\chi_{8008}(7597,\cdot)\)
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
\((6007,4005,3433,4369,4929)\) → \((1,-1,e\left(\frac{1}{3}\right),e\left(\frac{1}{10}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
| \( \chi_{ 8008 }(541, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) |
sage: chi.jacobi_sum(n)