from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5166, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,20,33]))
pari: [g,chi] = znchar(Mod(181,5166))
Basic properties
| Modulus: | \(5166\) | |
| 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 \(\chi_{287}(181,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5166.ew
\(\chi_{5166}(181,\cdot)\) \(\chi_{5166}(559,\cdot)\) \(\chi_{5166}(685,\cdot)\) \(\chi_{5166}(937,\cdot)\) \(\chi_{5166}(1441,\cdot)\) \(\chi_{5166}(1693,\cdot)\) \(\chi_{5166}(1819,\cdot)\) \(\chi_{5166}(2197,\cdot)\) \(\chi_{5166}(2449,\cdot)\) \(\chi_{5166}(3205,\cdot)\) \(\chi_{5166}(3457,\cdot)\) \(\chi_{5166}(3709,\cdot)\) \(\chi_{5166}(3835,\cdot)\) \(\chi_{5166}(4087,\cdot)\) \(\chi_{5166}(4339,\cdot)\) \(\chi_{5166}(5095,\cdot)\)
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
\((2297,2215,3655)\) → \((1,-1,e\left(\frac{33}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 5166 }(181, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)