from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6027, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,20,7]))
pari: [g,chi] = znchar(Mod(685,6027))
Basic properties
| Modulus: | \(6027\) | |
| Conductor: | \(287\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{287}(111,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6027.cr
\(\chi_{6027}(97,\cdot)\) \(\chi_{6027}(391,\cdot)\) \(\chi_{6027}(685,\cdot)\) \(\chi_{6027}(832,\cdot)\) \(\chi_{6027}(1126,\cdot)\) \(\chi_{6027}(1420,\cdot)\) \(\chi_{6027}(2302,\cdot)\) \(\chi_{6027}(2449,\cdot)\) \(\chi_{6027}(2596,\cdot)\) \(\chi_{6027}(3478,\cdot)\) \(\chi_{6027}(3625,\cdot)\) \(\chi_{6027}(3919,\cdot)\) \(\chi_{6027}(4066,\cdot)\) \(\chi_{6027}(4948,\cdot)\) \(\chi_{6027}(5095,\cdot)\) \(\chi_{6027}(5242,\cdot)\)
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: | 40.40.63172957949423116502957480067191906200305068755882825968063357506461803384975161.1 |
Values on generators
\((4019,493,2794)\) → \((1,-1,e\left(\frac{7}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 6027 }(685, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) |
sage: chi.jacobi_sum(n)