from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5600, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,35,38,20]))
pari: [g,chi] = znchar(Mod(13,5600))
Basic properties
| Modulus: | \(5600\) | |
| Conductor: | \(5600\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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 5600.gw
\(\chi_{5600}(13,\cdot)\) \(\chi_{5600}(517,\cdot)\) \(\chi_{5600}(573,\cdot)\) \(\chi_{5600}(1077,\cdot)\) \(\chi_{5600}(1133,\cdot)\) \(\chi_{5600}(1637,\cdot)\) \(\chi_{5600}(2197,\cdot)\) \(\chi_{5600}(2253,\cdot)\) \(\chi_{5600}(2813,\cdot)\) \(\chi_{5600}(3317,\cdot)\) \(\chi_{5600}(3373,\cdot)\) \(\chi_{5600}(3877,\cdot)\) \(\chi_{5600}(3933,\cdot)\) \(\chi_{5600}(4437,\cdot)\) \(\chi_{5600}(4997,\cdot)\) \(\chi_{5600}(5053,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((351,4901,5377,801)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{19}{20}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 5600 }(13, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) |
sage: chi.jacobi_sum(n)