from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(81225, base_ring=CyclotomicField(1710))
M = H._module
chi = DirichletCharacter(H, M([1140,1197,1135]))
pari: [g,chi] = znchar(Mod(34,81225))
Basic properties
| Modulus: | \(81225\) | |
| Conductor: | \(81225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1710\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 81225.nt
\(\chi_{81225}(34,\cdot)\) \(\chi_{81225}(364,\cdot)\) \(\chi_{81225}(679,\cdot)\) \(\chi_{81225}(754,\cdot)\) \(\chi_{81225}(889,\cdot)\) \(\chi_{81225}(979,\cdot)\) \(\chi_{81225}(1219,\cdot)\) \(\chi_{81225}(1534,\cdot)\)
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: | $\Q(\zeta_{855})$ |
| Fixed field: | Number field defined by a degree 1710 polynomial (not computed) |
Values on generators
\((36101,77977,48376)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{7}{10}\right),e\left(\frac{227}{342}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
| \( \chi_{ 81225 }(34, a) \) | \(-1\) | \(1\) | \(e\left(\frac{26}{855}\right)\) | \(e\left(\frac{52}{855}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{26}{285}\right)\) | \(e\left(\frac{54}{95}\right)\) | \(e\left(\frac{4}{855}\right)\) | \(e\left(\frac{1297}{1710}\right)\) | \(e\left(\frac{104}{855}\right)\) | \(e\left(\frac{181}{1710}\right)\) | \(e\left(\frac{512}{855}\right)\) |
sage: chi.jacobi_sum(n)