from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1476, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,10,17]))
pari: [g,chi] = znchar(Mod(143,1476))
Basic properties
| Modulus: | \(1476\) | |
| Conductor: | \(492\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{492}(143,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1476.bl
\(\chi_{1476}(143,\cdot)\) \(\chi_{1476}(251,\cdot)\) \(\chi_{1476}(323,\cdot)\) \(\chi_{1476}(431,\cdot)\) \(\chi_{1476}(863,\cdot)\) \(\chi_{1476}(935,\cdot)\) \(\chi_{1476}(1115,\cdot)\) \(\chi_{1476}(1187,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | 20.20.272085713140080346621846982691144795684864.1 |
Values on generators
\((739,821,1441)\) → \((-1,-1,e\left(\frac{17}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1476 }(143, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)