from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1911, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,19,21]))
pari: [g,chi] = znchar(Mod(38,1911))
Basic properties
| Modulus: | \(1911\) | |
| Conductor: | \(1911\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | 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 1911.dp
\(\chi_{1911}(38,\cdot)\) \(\chi_{1911}(194,\cdot)\) \(\chi_{1911}(311,\cdot)\) \(\chi_{1911}(467,\cdot)\) \(\chi_{1911}(584,\cdot)\) \(\chi_{1911}(740,\cdot)\) \(\chi_{1911}(857,\cdot)\) \(\chi_{1911}(1013,\cdot)\) \(\chi_{1911}(1130,\cdot)\) \(\chi_{1911}(1286,\cdot)\) \(\chi_{1911}(1559,\cdot)\) \(\chi_{1911}(1676,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.42.305425876345967117984813585685115495869514190060385043589428153059520623416150791007163797256182673.1 |
Values on generators
\((638,1522,1471)\) → \((-1,e\left(\frac{19}{42}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 1911 }(38, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{9}{14}\right)\) |
sage: chi.jacobi_sum(n)