from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2805, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,0,24,5]))
pari: [g,chi] = znchar(Mod(196,2805))
Basic properties
| Modulus: | \(2805\) | |
| Conductor: | \(187\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{187}(9,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2805.et
\(\chi_{2805}(196,\cdot)\) \(\chi_{2805}(586,\cdot)\) \(\chi_{2805}(631,\cdot)\) \(\chi_{2805}(841,\cdot)\) \(\chi_{2805}(916,\cdot)\) \(\chi_{2805}(961,\cdot)\) \(\chi_{2805}(1171,\cdot)\) \(\chi_{2805}(1351,\cdot)\) \(\chi_{2805}(1681,\cdot)\) \(\chi_{2805}(1906,\cdot)\) \(\chi_{2805}(2116,\cdot)\) \(\chi_{2805}(2161,\cdot)\) \(\chi_{2805}(2236,\cdot)\) \(\chi_{2805}(2446,\cdot)\) \(\chi_{2805}(2491,\cdot)\) \(\chi_{2805}(2671,\cdot)\)
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: | 40.40.24562817038400928776197921239227357886542077974183334844678041435576602047153.1 |
Values on generators
\((1871,562,1531,496)\) → \((1,1,e\left(\frac{3}{5}\right),e\left(\frac{1}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(19\) | \(23\) | \(26\) |
| \( \chi_{ 2805 }(196, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{20}\right)\) |
sage: chi.jacobi_sum(n)