from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2888, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,0,26]))
pari: [g,chi] = znchar(Mod(1103,2888))
Basic properties
| Modulus: | \(2888\) | |
| Conductor: | \(1444\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1444}(1103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2888.bd
\(\chi_{2888}(39,\cdot)\) \(\chi_{2888}(191,\cdot)\) \(\chi_{2888}(343,\cdot)\) \(\chi_{2888}(495,\cdot)\) \(\chi_{2888}(647,\cdot)\) \(\chi_{2888}(799,\cdot)\) \(\chi_{2888}(951,\cdot)\) \(\chi_{2888}(1103,\cdot)\) \(\chi_{2888}(1255,\cdot)\) \(\chi_{2888}(1407,\cdot)\) \(\chi_{2888}(1559,\cdot)\) \(\chi_{2888}(1711,\cdot)\) \(\chi_{2888}(1863,\cdot)\) \(\chi_{2888}(2015,\cdot)\) \(\chi_{2888}(2319,\cdot)\) \(\chi_{2888}(2471,\cdot)\) \(\chi_{2888}(2623,\cdot)\) \(\chi_{2888}(2775,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
| Fixed field: | 38.0.32314623763625504522847581826131926264699228491488831973421099549792171888378286207518194939116004573184.1 |
Values on generators
\((2167,1445,2529)\) → \((-1,1,e\left(\frac{13}{19}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 2888 }(1103, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) |
sage: chi.jacobi_sum(n)